- Email: [email protected]
Some new upper bounds of ex(n;{C3,C4})
Available online at www.sciencedirect.com
ScienceDirect AKCE International Journal of Graphs and Combinatorics (
)
– www.elsevier.com/locate/akcej
Some new upper bounds of ex(n; {C3, C4}) Novi H. Bong School of Electrical Engineering and Computer Science, University of Newcastle, Australia Received 4 May 2016; received in revised form 21 March 2017; accepted 21 March 2017 Available online xxxxx
Abstract The extremal number ex(n; {C3 , C4 }) or simply ex(n; 4) denotes the maximal number of edges in a graph on n vertices with forbidden subgraphs C3 and C4 . The exact number of ex(n; 4) is only known for n up to 32 and n = 50. There are upper and lower bounds of ex(n; 4) for other values of n. In this paper, we improve the upper bound of ex(n; 4) for n = 33, 34, . . . , 42 and also n = d 2 + 1 for any positive integer d ̸= 7, 57. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Extremal graph; Extremal number; Forbidden subgraph
1. Introduction Let G = (V, E) be a simple graph with vertex set V (G) and edge set E(G). The order and the size of a graph are defined to be the number of vertices and edges, respectively. The number of edges incident to a vertex v is called the degree of v (d(v)). If all vertices in G have the same degree r , then G is said to be regular or more specifically, r -regular. Let δ(∆) denote the minimum (maximum) degree of a graph and girth, g(G), denote the length of the smallest cycle in a graph. The distance between 2 vertices in a graph is defined to be the smallest number of edges connecting those two vertices. The maximum distance from a vertex to all other vertices is called the eccentricity of a vertex. The diameter, D, of a graph G is the maximum eccentricity over all the vertices in a graph. In this paper, we discuss the size maximality of graphs with some constraints. The graphs are required to have maximum number of edges without containing some given subgraphs. In general, let F be a family of graphs. A graph is called F-free if it does not contain any subgraph isomorphic to any of the graphs in F. The typical question that arises is: “How many edges can an F-free graph with n vertices have?” The maximum number of edges in an F-free graph on n vertices is denoted by ex(n; F). The family of graphs F itself is called “family of forbidden subgraphs”. There are several variations of F that have been considered, including Peer review under responsibility of Kalasalingam University. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.akcej.2017.03.006 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
2
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
triangles, cliques, squares, claw graphs (K 1,3 ), and the set of cycles of given lengths. The first forbidden subgraph considered was a triangle, F = {C3 }. Initiated by Mantel’s Theorem in 1907, if a graph G on n vertices contains more than n 2 /4 edges, then G contains a triangle. Hence, the extremal number of triangle-free graphs is at most ⌊n 2 /4⌋. This bound is obtained by the complete bipartite graphs K ⌊n/2⌋⌈n/2⌉ . In 1975, Erd˝os posed the problem of finding the maximum number of edges in graphs with n vertices that do not contain C3 or C4 . By ex(n; {C3 , C4 }), or simply ex(n; 4), we mean the maximum number of edges in a graph of order n and girth g ≥ 5. A graph that has ex(n; 4) edges is called an extremal graph. The set of all those extremal graphs is denoted by E X (n; 4). Erd˝os in [1] conjectured that ex(n; 4) = (1/2 + o(1))3/2 n 3/2 . In [2], Wang constructed regular graphs of degree d = 2k + 1 and n = 2d 2 − 4d + 2. The number of edges in these graphs also attains the best-known lower bound and asymptotically approaches the value in Erd˝os’ conjecture. Garnick et al. in [3] gave the exact value of ex(n; 4) for all n up to 24 and constructive lower bound for all n up to 200 by employing an algorithm involving both hill-climbing and backtracking techniques. They also enumerated all of the extremal graphs of order less than 21. Most of their results were verified by McKay’s Nauty program [4,5]. Additional values of ex(n; 4) for 25 ≤ n ≤ 30 were determined by √ Garnick and Nieuwejaar [6]. The upper bound for this problem is the following [7]: ex(n; {C3 , C4 }) ≤ n 2n−1 . In Section 2, we mention some properties of extremal graphs of girth 5 that will be useful in our proofs. In Section 3, we discuss the new upper bound of ex(n; 4), for n = 33, . . . , 42 and n = d 2 + 1 for any positive integer d ̸= 7, 57. In the end of this paper, we summarise the known exact values, the lower and the upper bound for the extremal number in Table 1. 2. Properties of extremal graphs The girth of a graph is related to its diameter by g ≤ 2D + 1. Since we are dealing with graphs of girth at least 5, then for n ≥ 3, the diameter is at least 2. The upper bound of the diameter of the extremal graphs is given in Proposition 2.1. Proposition 2.1 ([3]). Let G be an extremal {C3 , C4 }-free graph of order n. 1. The diameter of G is at most 3. 2. Suppose that the minimum degree of graph G is equal to 1 and let x be a vertex with degree 1, d(x) = δ(G) = 1, then the graph G − {x} has diameter at most 2. Some parameters of an extremal graph are related to its extremal number as stated in the following proposition. Proposition 2.2 ([3]). For all {C3 , C4 }-free graphs G of order n ≥ 1 and m edges, then 1. n ≥ 1 + ∆δ ≥ δ 2 ; ⌉; 2. δ ≥ m − ex(n − 1; 4) and ∆ ≥ ⌈ 2m n 3. n ≥ 1 + ⌈2ex(n; 4)/n⌉(ex(n; 4) − ex(n − 1; 4)). Proposition 2.2 shows that the knowledge of the extremal number of order n − 1 is indeed useful in determining the extremal number of order n. For example, it gives a bound for the degree. The second point in Proposition 2.2 says that if we have a graph with minimum degree δ and size m, then δ ≥ m − ex(n − 1; 4). This guarantees that removing a vertex with degree δ will not give a graph of size more than ex(n − 1; 4). Besides considering removing a single vertex from a graph, Garnick and Nieuwejaar considered the removal of a larger subgraph from the graph, as in Proposition 2.3. Proposition 2.3 ([6]). For any k vertices x1 , x2 , . . . , xk in a {C3 , C4 }-free graph G with order n > 1 and size m > 1, let d(xi ) be the degree of vertex xi and ⟨x1 , x2 , . . . , xk ⟩ be the vertex induced subgraph of G, then k ∑
d(xi ) − |E(⟨x1 , x2 , . . . , xk ⟩)| ≥ m − ex(n − k; 4)
i=1
where |E(⟨x1 , x2 , . . . , xk ⟩)| denotes the number of edges in the induced subgraph. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
3
√ Garnick et al. [3] proved that ex(n; 4) is bounded above by 21 n n − 1. This result implies Corollary 2.4. Corollary 2.4 ([3]). Let G be an extremal {C3 , C4 }-free graph of order n, size m, diameter 3 not regular, then √ 1 2 5 ex(n; 4) = m ≤ n (n − 1) − n. 2 2 In the beginning of this section, we mentioned that the girth of extremal graphs that do not contain C3 or C4 is at least 5. Garnick and Nieuwejaar [6] later on proved Theorem 2.5 which determined the girth of the graph. Theorem 2.5 ([6]). For all extremal graphs G, where n ≥ 5, and for all x ∈ V (G): 1. If d(x) = 2, then G has a C5 or C6 that contains x; 2. If d(x) ≥ 3, then G has a C5 that contains x. For n ≥ 7, it is trivial to construct a {C3 , C4 }-free graph with n vertices and n+1 edges. This means ex(n; 4) ≥ n+1 and the average degree of the vertices is greater than 2. Therefore, for any graph G ∈ E X (n; 4), n ≥ 7 there exists a vertex of degree at least 3. By Theorem 2.5, that particular vertex lies in a C5 . Hence, the girth of extremal graphs G that do not contain triangles and squares with n ≥ 7 is equal to 5 (g(G) = 5). Furthermore, an extremal graph with maximum degree ∆ ≥ t is necessarily of girth t + 1. Theorem 2.6 ([8]). Let t ≥ 3, G ∈ E X (n; t) and ∆(G) ≥ t, then g(G) = t + 1. 3. New upper bounds The exact values of extremal number ex(n; 4) are known up to n = 32. In general, the upper bound of ex(n; 4) can be computed using Proposition 2.2(3). The summary of the known exact values and the upper and lower bounds is given in [9]. The bound of ex(33; 4) is given by 87 ≤ ex(33; 4) ≤ 90 [9,10]. In Theorem 3.1, we prove that ex(33; 4) ̸= 90. Theorem 3.1. ex(33; 4) ≤ 89. Proof. Let G be the graph on 33 vertices having 90 edges. We observe the structure of G. 1. The minimum degree is 5 and the maximum degree is 6. 2 × 90 ≈ 5.45 ≤ ∆. (3.1) δ≤ 33 If δ ≤ 4, then deleting the vertex of degree δ gives us a graph of 32 vertices and 90 − δ ≥ 86 edges, however, from [9,10] we have ex(32; 4) = 85, thus 5 ≤ δ ≤ 5.45. Therefore, δ = 5. From Eq. (3.1), we have ∆ ≥ 6. To determine the exact value of ∆, we use the spanning tree of the graph. The spanning tree (from any vertex) must have distinct vertices in the first two levels, otherwise C3 or C4 will occur in the graph. If ∆ ≥ 7, then the spanning tree starting from a vertex of degree 7 must contain at least 1 + 7 + 7 × 4 = 36 distinct vertices. However, we only have 33 vertices. Thus, ∆ = 6. 2. There are 18 vertices of degree 5 and 15 vertices of degree 6 in G. Let x, y be the number of vertices with degree 5 and 6 respectively. Considering the number of vertices and also edges, we have two equations as follows: x + y = 33 and 5x + 6y = 180, with the solutions x = 18 and y = 15. 3. G cannot contain an adjacent pair of degree 5 vertices. Suppose G has an adjacent pair of degree 5 vertices, then applying Proposition 2.3, 2 ∑
d(xi ) − 1 = 2 × 5 − 1 = 9 < 10 = 90 − ex(31; 4).
i=1
This means, by deleting an adjacent pair of degree 5 vertices, we will obtain a graph of 31 vertices with 81 edges. This contradicts the fact that ex(31; 4) = 80 [9,10]. Hence, each vertex of degree 5 should be connected only to vertices of degree 6. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
4
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
From calculation, we have 18 vertices of degree 5, thus, they require 18 × 5 = 90 connections to vertices of degree 6. Since we have 15 vertices of degree 6 with exactly 90 connections, this forces all vertices of degree 6 to be connected to vertices of degree 5 only. Now, pick one vertex of degree 5 and consider the spanning tree from this vertex. In order to avoid triangles or squares, this spanning tree should contain all distinct vertices in the first 2 levels. Since the root is a vertex of degree 5, then at the first level, we need 5 vertices of degree 6. At the second level, we need 25 distinct vertices and all of them should be of degree 5. Unfortunately, from calculations, there are only 18 vertices of degree 5, which leads to a contradiction. Therefore, extremal graph of girth 5 on 33 vertices and 90 edges cannot exist. □ The bound of ex(34; 4) is given by 90 ≤ ex(34; 4) ≤ 94 [9]. Lemma 3.2. Assume ex(34; 4) = 94 and let G ∈ E X (34; 4) then 1. 2. 3. 4.
The minimum degree is 5 and the maximum degree is 6. There are 16 vertices of degree 5 and 18 vertices of degree 6 in G. G cannot contain a path of length 3 consisting only of degree 5 vertices. There are only 2 types of connections for vertices of degree 5, one that connects to 4 vertices of degree 6 and 1 vertex of degree 5 and one that connects to five vertices of degree 6. 5. Every vertex of degree 6 can be connected to at most 3 vertices of degree 6. 6. Every vertex of degree 6 must be connected to at least 1 other vertex of degree 6. Proof. Points 1, 2 and 5 are obtained in a similar way as in the proof of Theorem 3.1. 3. Deleting path P3 of degree 5 vertices will remove 13 edges, resulting in a graph on 31 vertices with 81 edges. This contradicts the fact that ex(31; 4) = 80 [10]. Hence, each vertex of degree 5 can be connected to at most 1 vertex of degree 5. This implies Point 4. 6. Suppose there exists a vertex u of degree 6 which is connected only to vertices of degree 5. The spanning tree starting from u will have 6 vertices of degree 5 at level 1. Based on Point 3 of this lemma, we need at least 6 × 3 = 18 distinct vertices of degree 6 at level 2. However, there are only 17 vertices of degree 6 excluding u that can be placed. □ The following lemma describes the structure of the possible breadth first search trees of depth 2 of the graph. Lemma 3.3. Assume ex(34; 4) = 94 and let G ∈ E X (34; 4). Take two vertices of degree 6 in G, u, v ∈ G, where u connects to exactly 1 vertex of degree 6 and v connects to exactly 3 vertices of degree 6. These vertices give rise to two unique non-isomorphic breadth first search trees of depth 2, one rooted at u as shown in Fig. 2 and one rooted at v as shown in Fig. 5. Proof. In what follows, by leaf we refer to a leaf of the breadth first search tree of depth 2. The graph itself has no leaves. 1. Consider the breadth first search tree of depth 2 starting from u. Every vertex of degree 5 at level 1 is connected to at least 3 vertices of degree 6. Therefore, there are two possibilities for the connections at level 2. (i) The vertex of degree 6 at level 1 connects to 5 vertices of degree 5 at level 2 (see Fig. 1). To complete the degree, all the leaves from the first branch (5 vertices of degree 5) must be connected to 5 × 4 = 20 distinct vertices. Note that these leaves cannot be connected to the leaf of degree 5 from the last 4 branches, since connecting them would result in a path P3 of vertices degree 5. Hence, there are at most 18 distinct vertices available for connections, which is impossible. (ii) The vertex of degree 6 at level 1 connects to 1 vertex of degree 6 and 4 vertices of degree 5. Since there are only 18 vertices of degree 6, in this case, there is only one possible breadth first search tree of depth 2 (shown in Fig. 2). 2. Consider the breadth first search tree of depth 2 starting from v. Vertex v reaches all other vertices within two steps. There are 4 possible arrangements for vertices at level 1. They are: Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
5
Fig. 1. The vertex of degree 6 at level 1 connects to 5 vertices of degree 5 at level 2.
Fig. 2. The vertex of degree 6 at level 1 connects to 1 vertex of degree 6 and 4 vertices of degree 5.
Fig. 3. Each vertex of degree 6 at level 1 connects to at least 1 vertex of degree 6 at level 2.
(i) There exists a vertex of degree 6 at level 1 that connects to 5 vertices of degree 5. To complete the degree, the 5 leaves of degree 5 must be connected to at least 15 distinct leaves of degree 6. However, we have only 18 vertices of degree 6 and at most 14 of them can be placed at level 2 as leaves. This is impossible. Hence, each vertex of degree 6 at level 1 connects to at least 1 vertex of degree 6 at level 2 (Fig. 3). (ii) All vertices of degree 6 at level 1 are connected to exactly 1 leaf of degree 6 at level 2 (Fig. 4). To complete the degree, the leaf of degree 6 from the first branch must be connected to exactly one vertex from every other branch. Thus, it is forced to connect to at least 2 vertices of degree 6 (one of each from the last two branches). We have 11 leaves of degree 6 remaining, while we still have 4 vertices of degree 5 from the first branch that have to be connected to at least 12 distinct vertices of degree 6. Hence, this breadth first search tree of depth 2 is impossible. (iii) Exactly one vertex of degree 6 at level 1 is connected to 2 leaves of degree 6. The other two vertices of degree 6 are connected to only 1 vertex of degree 6 at level 2. Using a similar argument as in (ii), this breadth first search tree of depth 2 is also impossible. (iv) Two vertices of degree 6 at level 1 are both connected to 2 leaves of degree 6 and 1 vertex of degree 6 is connected to 1 leaf of degree 6. This is the only possible breadth first search tree of depth 2 from vertex v and it is shown in Fig. 5. □ In Theorem 3.4, we prove that ex(34; 4) ̸= 94. Theorem 3.4. ex(34; 4) ≤ 93. Proof. Let G be the graph on 34 vertices having 94 edges. In Lemma 3.2, we know that there are two possible connections for vertices of degree 5. However, the connection of vertex of degree 5 with five vertices of degree 6 could not exist. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
6
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
Fig. 4. Each vertex of degree 6 at level 1 connects to exactly 1 vertex of degree 6 at level 2.
Fig. 5. Spanning tree starting from a vertex of degree 6 which is connected to 3 vertices of degree 6.
Fig. 6. Spanning tree starting from a vertex of degree 5 which is connected to all vertices of degree 6.
Fig. 7. The types of vertices in G.
Suppose we have a vertex of degree 5 that connects to 5 vertices of degree 6. Consider the vertices of degree 6 at level 1. In the breadth first search trees of depth 2 given by Figs. 2 and 5 in Lemma 3.3, we know that this type of vertex cannot exist. So, each vertex of degree 6 should be connected to exactly 2 vertices of degree 6. The breadth first search tree of depth 2 starting from a vertex of degree 5, that connects to 5 vertices of degree 6 can only be the one shown in Fig. 6. To complete the degree, each vertex of degree 6 at level 2, should connect to 1 of the 3 isolated vertices of degree 6. However, there are 10 vertices of degree 6 at level 2 and only 3 isolated vertices. By pigeonhole principle, one of the isolated vertices will have to connect to 4 vertices of degree 6, but we know from Lemma 3.2, that this is not possible. Hence, each vertex of degree 5 must be connected to exactly one vertex of degree 5. Recall the breadth first search tree of depth 2 starting from vertex u in Lemma 3.3 (Fig. 2). Consider those 4 vertices of degree 5 at level 2 that are connected to a vertex of degree 6 in the first branch. Each of them must be connected to 1 vertex of degree 5. Note that these vertices cannot be connected to vertices of degree 5 at level 2 since this will create the forbidden P3 . Therefore, these vertices must be connected to the 2 isolated vertices of degree 5. However, constructing 4 connections of this kind is impossible without creating C4 . This concludes that the breadth first search tree of depth 2 starting from vertex u cannot exist. Now, we know that each vertex of degree 6 must be connected to 2 or 3 vertices of degree 6. Suppose each vertex of degree 6 is connected to exactly 2 vertices of degree 6. Since there are 18 vertices of degree 6, then each vertex should have 4 connections to vertices of degree 5, thus, we need 4 × 18 = 72. However, each vertex of degree 5 has to be connected to 1 vertex of degree 5, so those 16 vertices only have 16 × 4 = 64 < 72 connections available. This arrangement is not possible. Therefore, there exists a vertex of degree 6 connecting to 3 vertices of degree 6. At this point, G has only 3 types of vertices, shown in Fig. 7. There are 16 vertices of type (a). Solving the equations 16 + 4b + 3c = 80 and 64 + 2b + 3c = 108 tells us there are 10 and 8 vertices of type (b) and (c) respectively. Further observation gives us: Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
7
Fig. 8. The only possible spanning tree starting from a vertex of degree 6 connecting to 3 vertices of degree 6.
Fig. 9. The breadth first search tree of depth 2 rooted at vertex 4.
1. A vertex of type (b) can be connected to at most 1 vertex of type (c). 2. A vertex of degree 5 (type (a)) must be connected to exactly 2 vertices of type (c). • A vertex of degree 5 (type (a)) must be connected to at most 2 vertices of type (c). Consider breadth first search tree of depth 2 starting from a degree 5 vertex. If this vertex connects to more than 2 vertices of type (c), the breadth first search tree of depth 2 requires more than 18 vertices of degree 6. • A vertex of degree 5 (type (a)) must be connected to at least 2 vertices of type (c). Otherwise, if we consider the breadth first search tree of depth 2 rooted at a vertex of degree 5, there will be a vertex of type (c) at distance 3 from the root. However, in the proof of Lemma 3.3, we know that the vertex of type (c) reaches all vertices in 2 steps, thus, a contradiction. Consider the unique breadth first search tree of depth 2 (up to isomorphism) starting from a vertex of type (c). The root reaches every vertex in two steps (Fig. 8). From spanning tree in Fig. 8, take vertex 4 as a new root and we obtain Fig. 9. This breadth first search tree of depth 2 requires 11 vertices of type (c), which is impossible since from calculation there are only 8 available. Therefore, the graph G on 34 vertices with 94 edges does not exist. □ As mentioned earlier that the knowledge of the extremal number of order n −1 is useful in determining the extremal number of order n, improving the bound for n − 1, to some extent, will also improve the bound of order n. Hence, the result in Theorem 3.4 implies the following corollary, by which we are able to improve the known bounds. Corollary 3.5. ex(35; 4) ≤ 98, ex(36; 4) ≤ 103 and ex(37; 4) ≤ 108. Proof. The known upper bound for ex(35; 4), ex(36; 4) and ex(37; 4) are 100, 105 and 111, respectively. Let m denote the number of edges in the graph. For n = 35, . . . , 37, the result is obtained by using the inequality in Proposition 2.2, m − ex(n − 1; 4) ≤ δ ≤ ⌊ 2m ⌋. □ n Theorem 3.6. ex(38; 4) ≤ 113 and ex(39; 4) ≤ 118. Proof. The current upper bound for ex(38; 4) is 115. By using the inequality in Proposition 2.2, m − ex(n − 1; 4) ≤ δ ≤ ⌊ 2m ⌋, it is easy to see that ex(38; 4) ̸= 115. n Furthermore, if ex(38; 4) = 114, 2 × 114 δ≤ = 6 ≤ ∆. 38 The value of δ cannot be 5 since ex(37; 4) ≤ 108 (Corollary 3.5), so δ = 6. This forces ∆ = 6. Hence, if this graph exists, it must be a 6-regular graph of girth 5 on 38 vertices. However, from the cages problem, the smallest 6-regular Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
8
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
graph of girth 5 ((6, 5)-cages) has 40 vertices [11] and its minimality is proven [12]. Thus, ex(38; 4) ≤ 113. For n = 39, if ex(39; 4) = 120, then 2 × 120 ≈ 6.15 ≤ ∆. 39 And if ex(39; 4) = 119, then δ≤
(3.2)
2 × 119 ≈ 6.1 ≤ ∆. (3.3) 39 From both Eqs. (3.2) and (3.3), we know that ∆ ≥ 7. If δ = 6, then the breadth first search tree of depth 2 starting from a vertex of maximum degree must contain at least 1 + 7 + 7 × 5 = 43 distinct vertices. This is impossible since there are only 39 vertices. So, δ ≤ 5. However, deleting a vertex of degree δ will give us a graph of 38 vertices with 115 and 114 edges, respectively. This contradicts the fact that ex(38; 4) ≤ 113. Hence, ex(39; 4) ≤ 118. □ δ≤
Similar to Theorem 3.4, the new bounds of ex(38; 4) and ex(39; 4) improve the bound on ex(n; 4) for n = 40, 41 and 42. Corollary 3.7. ex(40; 4) ≤ 123, ex(41; 4) ≤ 128 and ex(42; 4) ≤ 133. Besides improving the bounds of extremal number for consecutive n, we consider reducing the upper bounds of ex(n; 4) where n = d 2 + 1, for positive integers d. Theorem 3.8. For d ≥ 4, d ̸= 7, 57, ex(d 2 + 1; 4) <
d 3 +d . 2 3
Proof. Let G be the extremal graph on (d 2 + 1) vertices with d 2+d edges, where d 2 + 1 is the Moore bound of graphs with diameter 2. If this graph G exists, by Proposition 2.1, the diameter of G is at most 3. G cannot be a regular graph of diameter 2, since it will make G be a Moore graph of diameter 2, while Moore graphs of diameter 2 do not exist for d ̸∈ {2, 3, 7, 57} [13]. So, in this proof we only need to consider 3 cases. Case 1: G is a non-regular graph of diameter 2. This graph does not exist due to Singleton’s theorem that there is no irregular Moore graph [14]. An alternative proof for this case can be found in [15]. Case 2: G is a regular graph of diameter 3. Let G be a graph with d 2 + 1 vertices and diameter 3. Suppose that G is regular of degree ≥ d. Since the diameter is 3, then there is a pair of vertices, call them x, y, whose mutual distance is 3. Consider the spanning tree rooted at vertex x. If G is regular of degree ≥ d, then the spanning tree starting from x should contain at least d 2 + 1 distinct vertices within two levels. Hence, vertex x reaches all other vertices within at most two steps, which contradicts the fact that x and y have distance 3. Therefore, the degree of G should 2 be less than d. This implies that |E(G)| < d(d 2+1) . Case 3: G is a non-regular graph of diameter 3. Let G be a graph with n = d 2 + 1 vertices, size m, diameter 3 and not regular. Substituting n to the Corollary 2.4 [3], we obtain the upper bound for the number of edges. √ 1 5 (d 2 + 1)2 d 2 − (d 2 + 1) |E(G)| ≤ 2 2 ( ) 5 2 1√ 2 2 2 < (d + 1) d since (d + 1) > 0 2 2 d(d 2 + 1) = 2 In all cases, we show that it is impossible to have a graph G with d 2 + 1 vertices and case of regular graphs of diameter 2, i.e., when d = 7 or perhaps d = 57. □
d 3 +d 2
edges except for the
The summary of results is given in Table 1. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
0
0 15 41 76 120–123 175 207–230 257–290 320–355 388–424 455–497 512–574 591–654 665–738 734–825 825–915 923–1008 972–1104 1016–1204 1108–1306 1208–1410
n
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
0 16 44 80 124–128 176–180 212–236 263–297 326–362 395–431 459–504 520–582 599–662 668–746 743–834 834–924 934–1018 975–1114 1023–1214 1118–1316 1218–1421
1 1 18 47 85 129–133 178–185 216–242 269–303 332–368 402–438 462–512 528–589 607–670 672–755 752–843 843–933 945–1027 979–1124 1033–1224 1128–1326 1228–1431
2 2 21 50 87–89 134–139 181–191 221–248 275–309 339–375 409–446 466–520 536–597 615–679 675–764 761–852 853–943 948–1037 983–1134 1043–1234 1138–1337 1238–1442
3
Table 1 Updated values of exl (n; {C3 , C4 }) and exu (n; {C3 , C4 }), for n ≤ 210.
3 23 54 90–93 139–144 185–196 226–253 281–316 346–382 417–453 469–527 544–605 623–687 683–772 770–860 863–952 951–1046 987–1144 1052–1244 1148–1347 1248–1453
4 5 26 57 95–98 145–149 188–202 231–259 288–322 352–389 424–460 475–535 552–613 631–695 691–781 779–869 872–961 954–1056 992–1154 1061–1254 1158–1358 1258–1463
5 6 28 61 99–103 150–154 192–207 235–266 294–329 359–396 432–467 482–543 560–621 640–704 700–790 788–879 882–971 957–1066 997–1164 1071–1264 1168–1368 1268–1474
6 8 31 65 104–108 156–159 195–213 240–272 301–335 366–403 440–475 489–550 567–630 649–712 708–798 797–888 892–980 961–1075 1002–1174 1080–1275 1178–1378 1278–1485
7
10 34 68 109–113 162–164 199–218 245–278 307–342 373–410 449–482 497–558 575–638 658–721 717–807 806–897 902–989 964–1085 1006–1184 1089–1285 1188–1389 1288–1496
8
12 38 72 114–118 168–169 203–224 251–284 314–348 381–417 452–490 505–566 583–646 661–729 725–816 815–906 912–999 968–1095 1011–1194 1099–1295 1198–1400 1298–1507
9
N.H. Bong / AKCE International Journal of Graphs and Combinatorics ( ) – 9
Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
10
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
Acknowledgements The author would like to acknowledge Prof. Zdenˇek Ryj´acˇ ek for supporting her research visit at University of West Bohemia, Pilsen, Czech Republic, where part of this research was conducted. The author dedicates this paper to the memory of the author’s Ph.D. co-supervisor, Prof. Mirka Miller. References [1] P. Erd˝os, Some recent progress on extremal problems in graph theory, in: Proc. of the Sixth Southeastern Conf. on Combin., Graph Theory and Computing, Utilitas Math., Winnipeg, Man., 1975, pp. 3–14. No. XIV. [2] P. Wang, An upper bound for the (n, 5)-cages, Ars Combin. 47 (1997) 121–128. [3] D.K. Garnick, Y.H. Kwong, F. Lazebnik, Extremal graphs without three-cycles or four-cycles, J. Graph Theory 17 (5) (1993) 633–645. [4] B.D. McKay, http://cs.anu.edu.au/people/bdm. [5] B.D. McKay, Isomorph-free exhaustive generation, J. Algorithms 26 (2) (1998) 306–324. [6] D.K. Garnick, N.A. Nieuwejaar, Nonisomorphic extremal graphs without three-cycles or four-cycles, J. Combin. Math. Combin. Comput. 12 (1992) 33–56. [7] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001. [8] F. Lazebnik, P. Wang, On the structure of extremal graphs of high girth, J. Graph Theory 26 (3) (1997) 147–153. [9] C. Balbuena, K. Marshall, M. Miller, Contributions to the ex(n; {C3 , C4 }), preprint. [10] M. Codish, A. Miller, P. Prosser, P. Stuckey, Breaking symmetries in graph representation, in: IJCAI 2013, Proc. of the 23rd Int. Joint Conf. on Artificial Intelligence, IJCAI/AAAI, 2013. [11] M. O’Keefe, Pak Ken Wong, A smallest graph of girth 5 and valency 6, J. Combin. Theory Ser. B 26 (2) (1979) 145–149. [12] P.K. Wong, On the uniqueness of the smallest graph of girth 5 and valency 6, J. Graph Theory 3 (4) (1979) 407–409. [13] A.J. Hoffman, R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res. Develop. 4 (1960) 497–504. [14] R. Singleton, There is no irregular Moore graph, Amer. Math. Monthly 75 (1968) 42–43. [15] N.H. Bong, Properties and Structures of Extremal Graphs (Ph.D. thesis), The University of Newcastle, Australia, 2017.
Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
ScienceDirect AKCE International Journal of Graphs and Combinatorics (
)
– www.elsevier.com/locate/akcej
Some new upper bounds of ex(n; {C3, C4}) Novi H. Bong School of Electrical Engineering and Computer Science, University of Newcastle, Australia Received 4 May 2016; received in revised form 21 March 2017; accepted 21 March 2017 Available online xxxxx
Abstract The extremal number ex(n; {C3 , C4 }) or simply ex(n; 4) denotes the maximal number of edges in a graph on n vertices with forbidden subgraphs C3 and C4 . The exact number of ex(n; 4) is only known for n up to 32 and n = 50. There are upper and lower bounds of ex(n; 4) for other values of n. In this paper, we improve the upper bound of ex(n; 4) for n = 33, 34, . . . , 42 and also n = d 2 + 1 for any positive integer d ̸= 7, 57. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Extremal graph; Extremal number; Forbidden subgraph
1. Introduction Let G = (V, E) be a simple graph with vertex set V (G) and edge set E(G). The order and the size of a graph are defined to be the number of vertices and edges, respectively. The number of edges incident to a vertex v is called the degree of v (d(v)). If all vertices in G have the same degree r , then G is said to be regular or more specifically, r -regular. Let δ(∆) denote the minimum (maximum) degree of a graph and girth, g(G), denote the length of the smallest cycle in a graph. The distance between 2 vertices in a graph is defined to be the smallest number of edges connecting those two vertices. The maximum distance from a vertex to all other vertices is called the eccentricity of a vertex. The diameter, D, of a graph G is the maximum eccentricity over all the vertices in a graph. In this paper, we discuss the size maximality of graphs with some constraints. The graphs are required to have maximum number of edges without containing some given subgraphs. In general, let F be a family of graphs. A graph is called F-free if it does not contain any subgraph isomorphic to any of the graphs in F. The typical question that arises is: “How many edges can an F-free graph with n vertices have?” The maximum number of edges in an F-free graph on n vertices is denoted by ex(n; F). The family of graphs F itself is called “family of forbidden subgraphs”. There are several variations of F that have been considered, including Peer review under responsibility of Kalasalingam University. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.akcej.2017.03.006 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
2
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
triangles, cliques, squares, claw graphs (K 1,3 ), and the set of cycles of given lengths. The first forbidden subgraph considered was a triangle, F = {C3 }. Initiated by Mantel’s Theorem in 1907, if a graph G on n vertices contains more than n 2 /4 edges, then G contains a triangle. Hence, the extremal number of triangle-free graphs is at most ⌊n 2 /4⌋. This bound is obtained by the complete bipartite graphs K ⌊n/2⌋⌈n/2⌉ . In 1975, Erd˝os posed the problem of finding the maximum number of edges in graphs with n vertices that do not contain C3 or C4 . By ex(n; {C3 , C4 }), or simply ex(n; 4), we mean the maximum number of edges in a graph of order n and girth g ≥ 5. A graph that has ex(n; 4) edges is called an extremal graph. The set of all those extremal graphs is denoted by E X (n; 4). Erd˝os in [1] conjectured that ex(n; 4) = (1/2 + o(1))3/2 n 3/2 . In [2], Wang constructed regular graphs of degree d = 2k + 1 and n = 2d 2 − 4d + 2. The number of edges in these graphs also attains the best-known lower bound and asymptotically approaches the value in Erd˝os’ conjecture. Garnick et al. in [3] gave the exact value of ex(n; 4) for all n up to 24 and constructive lower bound for all n up to 200 by employing an algorithm involving both hill-climbing and backtracking techniques. They also enumerated all of the extremal graphs of order less than 21. Most of their results were verified by McKay’s Nauty program [4,5]. Additional values of ex(n; 4) for 25 ≤ n ≤ 30 were determined by √ Garnick and Nieuwejaar [6]. The upper bound for this problem is the following [7]: ex(n; {C3 , C4 }) ≤ n 2n−1 . In Section 2, we mention some properties of extremal graphs of girth 5 that will be useful in our proofs. In Section 3, we discuss the new upper bound of ex(n; 4), for n = 33, . . . , 42 and n = d 2 + 1 for any positive integer d ̸= 7, 57. In the end of this paper, we summarise the known exact values, the lower and the upper bound for the extremal number in Table 1. 2. Properties of extremal graphs The girth of a graph is related to its diameter by g ≤ 2D + 1. Since we are dealing with graphs of girth at least 5, then for n ≥ 3, the diameter is at least 2. The upper bound of the diameter of the extremal graphs is given in Proposition 2.1. Proposition 2.1 ([3]). Let G be an extremal {C3 , C4 }-free graph of order n. 1. The diameter of G is at most 3. 2. Suppose that the minimum degree of graph G is equal to 1 and let x be a vertex with degree 1, d(x) = δ(G) = 1, then the graph G − {x} has diameter at most 2. Some parameters of an extremal graph are related to its extremal number as stated in the following proposition. Proposition 2.2 ([3]). For all {C3 , C4 }-free graphs G of order n ≥ 1 and m edges, then 1. n ≥ 1 + ∆δ ≥ δ 2 ; ⌉; 2. δ ≥ m − ex(n − 1; 4) and ∆ ≥ ⌈ 2m n 3. n ≥ 1 + ⌈2ex(n; 4)/n⌉(ex(n; 4) − ex(n − 1; 4)). Proposition 2.2 shows that the knowledge of the extremal number of order n − 1 is indeed useful in determining the extremal number of order n. For example, it gives a bound for the degree. The second point in Proposition 2.2 says that if we have a graph with minimum degree δ and size m, then δ ≥ m − ex(n − 1; 4). This guarantees that removing a vertex with degree δ will not give a graph of size more than ex(n − 1; 4). Besides considering removing a single vertex from a graph, Garnick and Nieuwejaar considered the removal of a larger subgraph from the graph, as in Proposition 2.3. Proposition 2.3 ([6]). For any k vertices x1 , x2 , . . . , xk in a {C3 , C4 }-free graph G with order n > 1 and size m > 1, let d(xi ) be the degree of vertex xi and ⟨x1 , x2 , . . . , xk ⟩ be the vertex induced subgraph of G, then k ∑
d(xi ) − |E(⟨x1 , x2 , . . . , xk ⟩)| ≥ m − ex(n − k; 4)
i=1
where |E(⟨x1 , x2 , . . . , xk ⟩)| denotes the number of edges in the induced subgraph. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
3
√ Garnick et al. [3] proved that ex(n; 4) is bounded above by 21 n n − 1. This result implies Corollary 2.4. Corollary 2.4 ([3]). Let G be an extremal {C3 , C4 }-free graph of order n, size m, diameter 3 not regular, then √ 1 2 5 ex(n; 4) = m ≤ n (n − 1) − n. 2 2 In the beginning of this section, we mentioned that the girth of extremal graphs that do not contain C3 or C4 is at least 5. Garnick and Nieuwejaar [6] later on proved Theorem 2.5 which determined the girth of the graph. Theorem 2.5 ([6]). For all extremal graphs G, where n ≥ 5, and for all x ∈ V (G): 1. If d(x) = 2, then G has a C5 or C6 that contains x; 2. If d(x) ≥ 3, then G has a C5 that contains x. For n ≥ 7, it is trivial to construct a {C3 , C4 }-free graph with n vertices and n+1 edges. This means ex(n; 4) ≥ n+1 and the average degree of the vertices is greater than 2. Therefore, for any graph G ∈ E X (n; 4), n ≥ 7 there exists a vertex of degree at least 3. By Theorem 2.5, that particular vertex lies in a C5 . Hence, the girth of extremal graphs G that do not contain triangles and squares with n ≥ 7 is equal to 5 (g(G) = 5). Furthermore, an extremal graph with maximum degree ∆ ≥ t is necessarily of girth t + 1. Theorem 2.6 ([8]). Let t ≥ 3, G ∈ E X (n; t) and ∆(G) ≥ t, then g(G) = t + 1. 3. New upper bounds The exact values of extremal number ex(n; 4) are known up to n = 32. In general, the upper bound of ex(n; 4) can be computed using Proposition 2.2(3). The summary of the known exact values and the upper and lower bounds is given in [9]. The bound of ex(33; 4) is given by 87 ≤ ex(33; 4) ≤ 90 [9,10]. In Theorem 3.1, we prove that ex(33; 4) ̸= 90. Theorem 3.1. ex(33; 4) ≤ 89. Proof. Let G be the graph on 33 vertices having 90 edges. We observe the structure of G. 1. The minimum degree is 5 and the maximum degree is 6. 2 × 90 ≈ 5.45 ≤ ∆. (3.1) δ≤ 33 If δ ≤ 4, then deleting the vertex of degree δ gives us a graph of 32 vertices and 90 − δ ≥ 86 edges, however, from [9,10] we have ex(32; 4) = 85, thus 5 ≤ δ ≤ 5.45. Therefore, δ = 5. From Eq. (3.1), we have ∆ ≥ 6. To determine the exact value of ∆, we use the spanning tree of the graph. The spanning tree (from any vertex) must have distinct vertices in the first two levels, otherwise C3 or C4 will occur in the graph. If ∆ ≥ 7, then the spanning tree starting from a vertex of degree 7 must contain at least 1 + 7 + 7 × 4 = 36 distinct vertices. However, we only have 33 vertices. Thus, ∆ = 6. 2. There are 18 vertices of degree 5 and 15 vertices of degree 6 in G. Let x, y be the number of vertices with degree 5 and 6 respectively. Considering the number of vertices and also edges, we have two equations as follows: x + y = 33 and 5x + 6y = 180, with the solutions x = 18 and y = 15. 3. G cannot contain an adjacent pair of degree 5 vertices. Suppose G has an adjacent pair of degree 5 vertices, then applying Proposition 2.3, 2 ∑
d(xi ) − 1 = 2 × 5 − 1 = 9 < 10 = 90 − ex(31; 4).
i=1
This means, by deleting an adjacent pair of degree 5 vertices, we will obtain a graph of 31 vertices with 81 edges. This contradicts the fact that ex(31; 4) = 80 [9,10]. Hence, each vertex of degree 5 should be connected only to vertices of degree 6. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
4
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
From calculation, we have 18 vertices of degree 5, thus, they require 18 × 5 = 90 connections to vertices of degree 6. Since we have 15 vertices of degree 6 with exactly 90 connections, this forces all vertices of degree 6 to be connected to vertices of degree 5 only. Now, pick one vertex of degree 5 and consider the spanning tree from this vertex. In order to avoid triangles or squares, this spanning tree should contain all distinct vertices in the first 2 levels. Since the root is a vertex of degree 5, then at the first level, we need 5 vertices of degree 6. At the second level, we need 25 distinct vertices and all of them should be of degree 5. Unfortunately, from calculations, there are only 18 vertices of degree 5, which leads to a contradiction. Therefore, extremal graph of girth 5 on 33 vertices and 90 edges cannot exist. □ The bound of ex(34; 4) is given by 90 ≤ ex(34; 4) ≤ 94 [9]. Lemma 3.2. Assume ex(34; 4) = 94 and let G ∈ E X (34; 4) then 1. 2. 3. 4.
The minimum degree is 5 and the maximum degree is 6. There are 16 vertices of degree 5 and 18 vertices of degree 6 in G. G cannot contain a path of length 3 consisting only of degree 5 vertices. There are only 2 types of connections for vertices of degree 5, one that connects to 4 vertices of degree 6 and 1 vertex of degree 5 and one that connects to five vertices of degree 6. 5. Every vertex of degree 6 can be connected to at most 3 vertices of degree 6. 6. Every vertex of degree 6 must be connected to at least 1 other vertex of degree 6. Proof. Points 1, 2 and 5 are obtained in a similar way as in the proof of Theorem 3.1. 3. Deleting path P3 of degree 5 vertices will remove 13 edges, resulting in a graph on 31 vertices with 81 edges. This contradicts the fact that ex(31; 4) = 80 [10]. Hence, each vertex of degree 5 can be connected to at most 1 vertex of degree 5. This implies Point 4. 6. Suppose there exists a vertex u of degree 6 which is connected only to vertices of degree 5. The spanning tree starting from u will have 6 vertices of degree 5 at level 1. Based on Point 3 of this lemma, we need at least 6 × 3 = 18 distinct vertices of degree 6 at level 2. However, there are only 17 vertices of degree 6 excluding u that can be placed. □ The following lemma describes the structure of the possible breadth first search trees of depth 2 of the graph. Lemma 3.3. Assume ex(34; 4) = 94 and let G ∈ E X (34; 4). Take two vertices of degree 6 in G, u, v ∈ G, where u connects to exactly 1 vertex of degree 6 and v connects to exactly 3 vertices of degree 6. These vertices give rise to two unique non-isomorphic breadth first search trees of depth 2, one rooted at u as shown in Fig. 2 and one rooted at v as shown in Fig. 5. Proof. In what follows, by leaf we refer to a leaf of the breadth first search tree of depth 2. The graph itself has no leaves. 1. Consider the breadth first search tree of depth 2 starting from u. Every vertex of degree 5 at level 1 is connected to at least 3 vertices of degree 6. Therefore, there are two possibilities for the connections at level 2. (i) The vertex of degree 6 at level 1 connects to 5 vertices of degree 5 at level 2 (see Fig. 1). To complete the degree, all the leaves from the first branch (5 vertices of degree 5) must be connected to 5 × 4 = 20 distinct vertices. Note that these leaves cannot be connected to the leaf of degree 5 from the last 4 branches, since connecting them would result in a path P3 of vertices degree 5. Hence, there are at most 18 distinct vertices available for connections, which is impossible. (ii) The vertex of degree 6 at level 1 connects to 1 vertex of degree 6 and 4 vertices of degree 5. Since there are only 18 vertices of degree 6, in this case, there is only one possible breadth first search tree of depth 2 (shown in Fig. 2). 2. Consider the breadth first search tree of depth 2 starting from v. Vertex v reaches all other vertices within two steps. There are 4 possible arrangements for vertices at level 1. They are: Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
5
Fig. 1. The vertex of degree 6 at level 1 connects to 5 vertices of degree 5 at level 2.
Fig. 2. The vertex of degree 6 at level 1 connects to 1 vertex of degree 6 and 4 vertices of degree 5.
Fig. 3. Each vertex of degree 6 at level 1 connects to at least 1 vertex of degree 6 at level 2.
(i) There exists a vertex of degree 6 at level 1 that connects to 5 vertices of degree 5. To complete the degree, the 5 leaves of degree 5 must be connected to at least 15 distinct leaves of degree 6. However, we have only 18 vertices of degree 6 and at most 14 of them can be placed at level 2 as leaves. This is impossible. Hence, each vertex of degree 6 at level 1 connects to at least 1 vertex of degree 6 at level 2 (Fig. 3). (ii) All vertices of degree 6 at level 1 are connected to exactly 1 leaf of degree 6 at level 2 (Fig. 4). To complete the degree, the leaf of degree 6 from the first branch must be connected to exactly one vertex from every other branch. Thus, it is forced to connect to at least 2 vertices of degree 6 (one of each from the last two branches). We have 11 leaves of degree 6 remaining, while we still have 4 vertices of degree 5 from the first branch that have to be connected to at least 12 distinct vertices of degree 6. Hence, this breadth first search tree of depth 2 is impossible. (iii) Exactly one vertex of degree 6 at level 1 is connected to 2 leaves of degree 6. The other two vertices of degree 6 are connected to only 1 vertex of degree 6 at level 2. Using a similar argument as in (ii), this breadth first search tree of depth 2 is also impossible. (iv) Two vertices of degree 6 at level 1 are both connected to 2 leaves of degree 6 and 1 vertex of degree 6 is connected to 1 leaf of degree 6. This is the only possible breadth first search tree of depth 2 from vertex v and it is shown in Fig. 5. □ In Theorem 3.4, we prove that ex(34; 4) ̸= 94. Theorem 3.4. ex(34; 4) ≤ 93. Proof. Let G be the graph on 34 vertices having 94 edges. In Lemma 3.2, we know that there are two possible connections for vertices of degree 5. However, the connection of vertex of degree 5 with five vertices of degree 6 could not exist. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
6
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
Fig. 4. Each vertex of degree 6 at level 1 connects to exactly 1 vertex of degree 6 at level 2.
Fig. 5. Spanning tree starting from a vertex of degree 6 which is connected to 3 vertices of degree 6.
Fig. 6. Spanning tree starting from a vertex of degree 5 which is connected to all vertices of degree 6.
Fig. 7. The types of vertices in G.
Suppose we have a vertex of degree 5 that connects to 5 vertices of degree 6. Consider the vertices of degree 6 at level 1. In the breadth first search trees of depth 2 given by Figs. 2 and 5 in Lemma 3.3, we know that this type of vertex cannot exist. So, each vertex of degree 6 should be connected to exactly 2 vertices of degree 6. The breadth first search tree of depth 2 starting from a vertex of degree 5, that connects to 5 vertices of degree 6 can only be the one shown in Fig. 6. To complete the degree, each vertex of degree 6 at level 2, should connect to 1 of the 3 isolated vertices of degree 6. However, there are 10 vertices of degree 6 at level 2 and only 3 isolated vertices. By pigeonhole principle, one of the isolated vertices will have to connect to 4 vertices of degree 6, but we know from Lemma 3.2, that this is not possible. Hence, each vertex of degree 5 must be connected to exactly one vertex of degree 5. Recall the breadth first search tree of depth 2 starting from vertex u in Lemma 3.3 (Fig. 2). Consider those 4 vertices of degree 5 at level 2 that are connected to a vertex of degree 6 in the first branch. Each of them must be connected to 1 vertex of degree 5. Note that these vertices cannot be connected to vertices of degree 5 at level 2 since this will create the forbidden P3 . Therefore, these vertices must be connected to the 2 isolated vertices of degree 5. However, constructing 4 connections of this kind is impossible without creating C4 . This concludes that the breadth first search tree of depth 2 starting from vertex u cannot exist. Now, we know that each vertex of degree 6 must be connected to 2 or 3 vertices of degree 6. Suppose each vertex of degree 6 is connected to exactly 2 vertices of degree 6. Since there are 18 vertices of degree 6, then each vertex should have 4 connections to vertices of degree 5, thus, we need 4 × 18 = 72. However, each vertex of degree 5 has to be connected to 1 vertex of degree 5, so those 16 vertices only have 16 × 4 = 64 < 72 connections available. This arrangement is not possible. Therefore, there exists a vertex of degree 6 connecting to 3 vertices of degree 6. At this point, G has only 3 types of vertices, shown in Fig. 7. There are 16 vertices of type (a). Solving the equations 16 + 4b + 3c = 80 and 64 + 2b + 3c = 108 tells us there are 10 and 8 vertices of type (b) and (c) respectively. Further observation gives us: Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
7
Fig. 8. The only possible spanning tree starting from a vertex of degree 6 connecting to 3 vertices of degree 6.
Fig. 9. The breadth first search tree of depth 2 rooted at vertex 4.
1. A vertex of type (b) can be connected to at most 1 vertex of type (c). 2. A vertex of degree 5 (type (a)) must be connected to exactly 2 vertices of type (c). • A vertex of degree 5 (type (a)) must be connected to at most 2 vertices of type (c). Consider breadth first search tree of depth 2 starting from a degree 5 vertex. If this vertex connects to more than 2 vertices of type (c), the breadth first search tree of depth 2 requires more than 18 vertices of degree 6. • A vertex of degree 5 (type (a)) must be connected to at least 2 vertices of type (c). Otherwise, if we consider the breadth first search tree of depth 2 rooted at a vertex of degree 5, there will be a vertex of type (c) at distance 3 from the root. However, in the proof of Lemma 3.3, we know that the vertex of type (c) reaches all vertices in 2 steps, thus, a contradiction. Consider the unique breadth first search tree of depth 2 (up to isomorphism) starting from a vertex of type (c). The root reaches every vertex in two steps (Fig. 8). From spanning tree in Fig. 8, take vertex 4 as a new root and we obtain Fig. 9. This breadth first search tree of depth 2 requires 11 vertices of type (c), which is impossible since from calculation there are only 8 available. Therefore, the graph G on 34 vertices with 94 edges does not exist. □ As mentioned earlier that the knowledge of the extremal number of order n −1 is useful in determining the extremal number of order n, improving the bound for n − 1, to some extent, will also improve the bound of order n. Hence, the result in Theorem 3.4 implies the following corollary, by which we are able to improve the known bounds. Corollary 3.5. ex(35; 4) ≤ 98, ex(36; 4) ≤ 103 and ex(37; 4) ≤ 108. Proof. The known upper bound for ex(35; 4), ex(36; 4) and ex(37; 4) are 100, 105 and 111, respectively. Let m denote the number of edges in the graph. For n = 35, . . . , 37, the result is obtained by using the inequality in Proposition 2.2, m − ex(n − 1; 4) ≤ δ ≤ ⌊ 2m ⌋. □ n Theorem 3.6. ex(38; 4) ≤ 113 and ex(39; 4) ≤ 118. Proof. The current upper bound for ex(38; 4) is 115. By using the inequality in Proposition 2.2, m − ex(n − 1; 4) ≤ δ ≤ ⌊ 2m ⌋, it is easy to see that ex(38; 4) ̸= 115. n Furthermore, if ex(38; 4) = 114, 2 × 114 δ≤ = 6 ≤ ∆. 38 The value of δ cannot be 5 since ex(37; 4) ≤ 108 (Corollary 3.5), so δ = 6. This forces ∆ = 6. Hence, if this graph exists, it must be a 6-regular graph of girth 5 on 38 vertices. However, from the cages problem, the smallest 6-regular Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
8
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
graph of girth 5 ((6, 5)-cages) has 40 vertices [11] and its minimality is proven [12]. Thus, ex(38; 4) ≤ 113. For n = 39, if ex(39; 4) = 120, then 2 × 120 ≈ 6.15 ≤ ∆. 39 And if ex(39; 4) = 119, then δ≤
(3.2)
2 × 119 ≈ 6.1 ≤ ∆. (3.3) 39 From both Eqs. (3.2) and (3.3), we know that ∆ ≥ 7. If δ = 6, then the breadth first search tree of depth 2 starting from a vertex of maximum degree must contain at least 1 + 7 + 7 × 5 = 43 distinct vertices. This is impossible since there are only 39 vertices. So, δ ≤ 5. However, deleting a vertex of degree δ will give us a graph of 38 vertices with 115 and 114 edges, respectively. This contradicts the fact that ex(38; 4) ≤ 113. Hence, ex(39; 4) ≤ 118. □ δ≤
Similar to Theorem 3.4, the new bounds of ex(38; 4) and ex(39; 4) improve the bound on ex(n; 4) for n = 40, 41 and 42. Corollary 3.7. ex(40; 4) ≤ 123, ex(41; 4) ≤ 128 and ex(42; 4) ≤ 133. Besides improving the bounds of extremal number for consecutive n, we consider reducing the upper bounds of ex(n; 4) where n = d 2 + 1, for positive integers d. Theorem 3.8. For d ≥ 4, d ̸= 7, 57, ex(d 2 + 1; 4) <
d 3 +d . 2 3
Proof. Let G be the extremal graph on (d 2 + 1) vertices with d 2+d edges, where d 2 + 1 is the Moore bound of graphs with diameter 2. If this graph G exists, by Proposition 2.1, the diameter of G is at most 3. G cannot be a regular graph of diameter 2, since it will make G be a Moore graph of diameter 2, while Moore graphs of diameter 2 do not exist for d ̸∈ {2, 3, 7, 57} [13]. So, in this proof we only need to consider 3 cases. Case 1: G is a non-regular graph of diameter 2. This graph does not exist due to Singleton’s theorem that there is no irregular Moore graph [14]. An alternative proof for this case can be found in [15]. Case 2: G is a regular graph of diameter 3. Let G be a graph with d 2 + 1 vertices and diameter 3. Suppose that G is regular of degree ≥ d. Since the diameter is 3, then there is a pair of vertices, call them x, y, whose mutual distance is 3. Consider the spanning tree rooted at vertex x. If G is regular of degree ≥ d, then the spanning tree starting from x should contain at least d 2 + 1 distinct vertices within two levels. Hence, vertex x reaches all other vertices within at most two steps, which contradicts the fact that x and y have distance 3. Therefore, the degree of G should 2 be less than d. This implies that |E(G)| < d(d 2+1) . Case 3: G is a non-regular graph of diameter 3. Let G be a graph with n = d 2 + 1 vertices, size m, diameter 3 and not regular. Substituting n to the Corollary 2.4 [3], we obtain the upper bound for the number of edges. √ 1 5 (d 2 + 1)2 d 2 − (d 2 + 1) |E(G)| ≤ 2 2 ( ) 5 2 1√ 2 2 2 < (d + 1) d since (d + 1) > 0 2 2 d(d 2 + 1) = 2 In all cases, we show that it is impossible to have a graph G with d 2 + 1 vertices and case of regular graphs of diameter 2, i.e., when d = 7 or perhaps d = 57. □
d 3 +d 2
edges except for the
The summary of results is given in Table 1. Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
0
0 15 41 76 120–123 175 207–230 257–290 320–355 388–424 455–497 512–574 591–654 665–738 734–825 825–915 923–1008 972–1104 1016–1204 1108–1306 1208–1410
n
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
0 16 44 80 124–128 176–180 212–236 263–297 326–362 395–431 459–504 520–582 599–662 668–746 743–834 834–924 934–1018 975–1114 1023–1214 1118–1316 1218–1421
1 1 18 47 85 129–133 178–185 216–242 269–303 332–368 402–438 462–512 528–589 607–670 672–755 752–843 843–933 945–1027 979–1124 1033–1224 1128–1326 1228–1431
2 2 21 50 87–89 134–139 181–191 221–248 275–309 339–375 409–446 466–520 536–597 615–679 675–764 761–852 853–943 948–1037 983–1134 1043–1234 1138–1337 1238–1442
3
Table 1 Updated values of exl (n; {C3 , C4 }) and exu (n; {C3 , C4 }), for n ≤ 210.
3 23 54 90–93 139–144 185–196 226–253 281–316 346–382 417–453 469–527 544–605 623–687 683–772 770–860 863–952 951–1046 987–1144 1052–1244 1148–1347 1248–1453
4 5 26 57 95–98 145–149 188–202 231–259 288–322 352–389 424–460 475–535 552–613 631–695 691–781 779–869 872–961 954–1056 992–1154 1061–1254 1158–1358 1258–1463
5 6 28 61 99–103 150–154 192–207 235–266 294–329 359–396 432–467 482–543 560–621 640–704 700–790 788–879 882–971 957–1066 997–1164 1071–1264 1168–1368 1268–1474
6 8 31 65 104–108 156–159 195–213 240–272 301–335 366–403 440–475 489–550 567–630 649–712 708–798 797–888 892–980 961–1075 1002–1174 1080–1275 1178–1378 1278–1485
7
10 34 68 109–113 162–164 199–218 245–278 307–342 373–410 449–482 497–558 575–638 658–721 717–807 806–897 902–989 964–1085 1006–1184 1089–1285 1188–1389 1288–1496
8
12 38 72 114–118 168–169 203–224 251–284 314–348 381–417 452–490 505–566 583–646 661–729 725–816 815–906 912–999 968–1095 1011–1194 1099–1295 1198–1400 1298–1507
9
N.H. Bong / AKCE International Journal of Graphs and Combinatorics ( ) – 9
Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.
10
N.H. Bong / AKCE International Journal of Graphs and Combinatorics (
)
–
Acknowledgements The author would like to acknowledge Prof. Zdenˇek Ryj´acˇ ek for supporting her research visit at University of West Bohemia, Pilsen, Czech Republic, where part of this research was conducted. The author dedicates this paper to the memory of the author’s Ph.D. co-supervisor, Prof. Mirka Miller. References [1] P. Erd˝os, Some recent progress on extremal problems in graph theory, in: Proc. of the Sixth Southeastern Conf. on Combin., Graph Theory and Computing, Utilitas Math., Winnipeg, Man., 1975, pp. 3–14. No. XIV. [2] P. Wang, An upper bound for the (n, 5)-cages, Ars Combin. 47 (1997) 121–128. [3] D.K. Garnick, Y.H. Kwong, F. Lazebnik, Extremal graphs without three-cycles or four-cycles, J. Graph Theory 17 (5) (1993) 633–645. [4] B.D. McKay, http://cs.anu.edu.au/people/bdm. [5] B.D. McKay, Isomorph-free exhaustive generation, J. Algorithms 26 (2) (1998) 306–324. [6] D.K. Garnick, N.A. Nieuwejaar, Nonisomorphic extremal graphs without three-cycles or four-cycles, J. Combin. Math. Combin. Comput. 12 (1992) 33–56. [7] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001. [8] F. Lazebnik, P. Wang, On the structure of extremal graphs of high girth, J. Graph Theory 26 (3) (1997) 147–153. [9] C. Balbuena, K. Marshall, M. Miller, Contributions to the ex(n; {C3 , C4 }), preprint. [10] M. Codish, A. Miller, P. Prosser, P. Stuckey, Breaking symmetries in graph representation, in: IJCAI 2013, Proc. of the 23rd Int. Joint Conf. on Artificial Intelligence, IJCAI/AAAI, 2013. [11] M. O’Keefe, Pak Ken Wong, A smallest graph of girth 5 and valency 6, J. Combin. Theory Ser. B 26 (2) (1979) 145–149. [12] P.K. Wong, On the uniqueness of the smallest graph of girth 5 and valency 6, J. Graph Theory 3 (4) (1979) 407–409. [13] A.J. Hoffman, R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res. Develop. 4 (1960) 497–504. [14] R. Singleton, There is no irregular Moore graph, Amer. Math. Monthly 75 (1968) 42–43. [15] N.H. Bong, Properties and Structures of Extremal Graphs (Ph.D. thesis), The University of Newcastle, Australia, 2017.
Please cite this article in press as: N.H. Bong, Some new upper bounds of ex(n; {C3 , C4 }), AKCE International Journal of Graphs and Combinatorics (2017), http://dx.doi.org/10.1016/j.akcej.2017.03.006.