- Email: [email protected]
Extremal Graphs without Cycles of Length 8 or Less
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Electronic Notes in Discrete Mathematics 38 (2011) 615–620 www.elsevier.com/locate/endm
Extremal Graphs without Cycles of Length 8 or Less Kim Marshall 1 School of El. Eng. and Comp. Sc., University of Newcastle, Australia
Mirka Miller 2,3 School of El. Eng. and Comp. Sc., University of Newcastle, Australia Department of Mathematics, University of West Bohemia, Czech Republic Department of Informatics, King’s College London, UK
Joe Ryan 4 School of El. Eng. and Comp. Sc., University of Newcastle, Australia
Abstract Let ex(n; t) denote the maximum number of edges in a graph G having order n without cycles of length t or less. We prove ex(23; 8) = 28, ex(24; 8) = 20 and ex(25; 8) = 30. Furthermore, we present new lower and upper bounds for n ≤ 49 and the extremal numbers when known. Keywords: Extremal graph, extremal number, girth.
1
Email: [email protected] Email: [email protected] 3 This research was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme 4 Email: [email protected] 2
1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2011.10.003
616
1
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
Introduction
Let G be a finite, simple graph of order n = |V (G)|. The extremal number, denoted ex(n; t), is the maximum size m = |E(G)| of a graph having order n and containing no cycle Ck such that k ≤ t. We say that G is an extremal graph, G ∈ EX(n; t), if the number of edges in G is equal to ex(n; t). When the exact value of ex(n; t) is not yet known we use the notation exl (n; t) and exu (n; t) to indicate the lower and upper bounds for the value of ex(n; t).
2
Known Results
In 1940 Tur´an [4] asked: “How many edges must a graph contain that it should certainly have subgraphs of a prescribed structure?”. A class of problems, called Tur´an type problems, stem from this question. The problem of finding the value of the extremal number ex(n; t) belongs to this class of problems. 2 Mantel’s theorem determined ex(n; 3) = n4 . Graphs that attain this bound are the complete bipartite graphs K n2 n2 . In 1975 Erd˝os [5] posed the problem of finding ex(n; 4), that is, the extremal number for graphs that do not contain three-cycles or four-cycles. Lazebnik and Wang [7] along with Garnick et al. [6] provided a summary of results for this problem and constructive lower bounds of ex(n; 4) for n ≤ 200 and exact values for n ≤ 30 and n = 50. More recent work has focused on finding ex(n; t) for t ≥ 4. Abajo and Di´anez [2] found ex(n; t) for all n ≤ (16t − 15)/5. These results include the current known exact values and upper and lower bounds for ex(n; 7) and ex(n; 8) which are displayed in Table 1 and in italics within Table 2. Abajo and Di´anez [3] also obtained bounds and in some cases exact values of ex(n; t) for t ∈ {5, 6, 7}. Improved lower bounds on the extremal number were determined in [1] by the construction of dense graphs for t ∈ {4, 6, 7, 10, 11}. In general the problem of determining the extremal number and constructing extremal graphs is very hard. As a consequence, research activities focus on constructing denser graphs to increase the lower bounds of ex(n; t) and on non-existence proofs to decrease the upper bound.
3
New Results
We begin with a result concerning the lower bound for an infinite series of graphs on n = k 2 vertices having girth 9.
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
n
0
1
2
3
4
5
6
7
8
9
0
0
0
1
2
3
4
5
6
8
9
10
10
12
13
14
16
18
19
20
22
24
20
25
27
29
30
32
34
36
38
40
42
30
45
46
47
49
51
53
55
617
Table 1 Known values of ex(n; 7) for n ≤ 19 by [3].
Theorem 3.1 For k ≥ 3, exl (k 2 ; 8) = 32 k(k − 1). Proof. The proof is by construction. The construction is the subdivision of the complete graph Kk . These graphs include; the cycle C9 for k = 3 and the graphs drawn below for k ∈ {4, 5}. 2 These graphs are known to be extremal for k ≤ 4 (see Table 2). We would like to prove that our construction produces an extremal graph G ∈ EX(n; 8) for k = 5. In order to do this we require Proposition 3.2.
s s @ s s @ s s
s s
s s
s s
@ @s s @ s @s
Subdivision of K4
s sZ s sBBsZ Z s B Zs s s s Zs B s s BZ sZZ B s B Z B B Z B B s s s Z s Bs s Z BBs s sZBs
Subdivision of K5
Proposition 3.2(i) The function ex(n; t) is strictly increasing in n and nondecreasing in t, that is, ex(n; t) ≥ ex(n−1; t)+1 and ex(n; t) ≤ ex(n; t−1). (ii) Let d be the average degree then δ ≤ d ≤ d ≤ d ≤ Δ. (iii) Let G ∈ EX(n; t) and v ∈ G be such that deg(v) = δ. Let G = G − {v}. Then G has order n − 1, g ≥ t and we obtain ex(n; t) − δ ≤ ex(n − 1; t).
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
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(iv) Combining (i) and (ii) gives the following inequality. ex(n; t) − ex(n − 1; t) ≤ δ ≤
ex(n; t) ∗ 2 ex(n; t) ∗ 2 ≤ ≤Δ n n
We denote by TΔ,δ a tree of height 4 and root r such that deg(r) = Δ and every non-leaf vertex v ∈ V (TΔ,δ ) − {r} has deg(v) = δ. Note that all G ∈ EX(n; 8) contain TΔ,δ as a subgraph. Furthermore, since g ≥ 9 it follows that the vertices in TΔ,δ must all be distinct and edges exist only between parent and child vertices. Note that |V (TΔ,δ )| = 1+Δ+Δ(δ−1)+Δ(δ−1)2 +Δ(δ−1)3 and |E(TΔ,δ )| = Δ + Δ(δ − 1) + Δ(δ − 1)2 + Δ(δ − 1)3 . Let X = V (G − TΔ,δ ). Our results are summarised in Table 2. Previously known values of ex(n; 8) determined by [3] are shown in italics. The new results for the exact value of ex(n; t) are shown in bold text. New lower and upper bounds are given for 26 ≤ n ≤ 49. The table includes only values for n ≤ 49 but our construction can be used to determine lower bounds also for larger n. n
0
1
2
3
4
5
6
7
8
9
0
0
0
1
2
3
4
5
6
7
9
10
10
11
12
14
15
16
18
19
21
22
20
23
25
27
28
29
30
31-32
32-34
33-36
34-38
30
36-40
37-42
39-44
40-46
42-48
43-50
45-52
46-54
47-56
48-58
40
50-60
52-62
54-64
56-66
58-68
60-70
62-73
64-76
66-79
68-82
Table 2 New values of ex(n; 8), for n = 23, 24, 25, and exl (n; 8) and exu (n; 8), for n ≤ 49.
Theorem 3.3 Let G ∈ EX(23; 8). Then |E(G)| = ex(23; 8) = 28. Sketch of Proof. By adding one edge and one vertex to a graph from EX(22; 8) we get the constructive lower bound exl (23; 8) = ex(22; 8)+1 = 28. The upper bound is obtained by combining Proposition 3.2 (i) and results in Table 1, namely, exu (23; 8) = ex(23; 7) = 30. Therefore, 28 ≤ ex(23; 8) ≤ 30. The only values in this range that satisfy the following inequality from Proposition 3.2 (iv) are 28 and 29. ex(23; 8) − ex(22; 8) ≤ δ ≤
ex(23; 8) ∗ 2 ex(23; 8) ∗ 2 ≤ ≤Δ 23 23
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
619
Therefore, δ = 2 and Δ ≥ 3. Assume ex(23, 8) = 29. We know that Δ ≤ 6 since otherwise |V (T6,2 )| ≥ 25. Therefore, assume Δ = 5 then |V (T5,2 )| ≥ 21 which is okay. However, ex(23; 8) − ex(21, 8) = 29 − 25 = 4 means that removing two vertices from G must remove at least 4 edges of G. That is, in G no two vertices having degree 2 are neighbours. The minimum spanning tree for a graph with Δ ≥ 4, and the additional constraint of no two vertices of degree 2 being neighbours, has at least 25 vertices. Therefore, Δ = 3, and D = {312 , 211 }. However, constructing a tree with root r such that deg(r) = 3 or deg(r) = 2 and no two vertices having degree two are neighbours is not possible. Therefore, ex(23; 8) = 28. 2 Theorem 3.4 Let G ∈ EX(24; 8). Then |E(G)| = ex(24; 8) = 29. Sketch of Proof. By construction, exl (24; 8) = ex(23; 8)+1 = 29. The upper bound is obtained from considering exu (24; 8) = ex(24; 7) = 32. Therefore, 29 ≤ ex(24; 8) ≤ 32. The only values in this range that satisfy Proposition 3.2 (iv) are 29 and 30. Assume that ex(24; 8) = 30. Since ex(24; 8) − ex(21 : 8) = 30 − 25 = 5, there is no path P3 such that all vertices in the path have degree 2. That is, removing 3 vertices must remove at least 5 edges. The minimum spanning tree T5,2 with this additional constraint has 26 vertices. Therefore Δ ≤ 4. We assume Δ = 4 and consider the minimum spanning tree T4,2 of depth 4 which, with the addition forbidden path constraint has |V (T )| = 21, |E(T )| = 20 and degree sequence D = {41 , 34 , 28 , 18 }. From this basis we show that it is impossible to add 3 extra vertices and 10 edges without constructing a cycle of length 8 or less. This gives Δ ≤ 3 and degree sequence D = {312 , 212 }. Considering the twelve vertices of degree 3 as S4 stars, we now resolve the structure of the graph and the lengths of the cycles on which the degree 3 vertices sit. Within this structure, it is not possible to accommodate the required number of edges without constructing cycles smaller than 9. Therefore ex(24; 8) = 29. 2 Theorem 3.5 Let G ∈ EX(25; 8). Then |E(G)| = ex(25; 8) = 30. Sketch of Proof. By construction, exl (25; 8) = ex(24; 8)+1 = 30. The upper bound is obtained from considering exu (25; 8) = ex(25; 7) = 34. Therefore, 30 ≤ ex(25; 8) ≤ 34. The only values in this range that also satisfy Proposition 3.2 (iv) are 30 and 31. Assume that ex(25; 8) = 31. Since ex(25; 8) − ex(21; 8) = 31 − 25 = 6 we cannot have a path of length 3 in G where every vertex has degree 2. That
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K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
is, removing four vertices must remove at least 6 edges. As a consequence of structural properties created by this forbidden path and the number of vertices in |V (TΔ,δ )|, we find that Δ must be less than 4 to avoid small cycles in G. By similar reasoning to Theorem 3.4 we find that Δ = 3 and D = {312 , 213 }. We reason as in Theorem 3.4 by considering the structure forced on the graph by the constraints and show that there must exist a cycle of length less than 9. Therefore ex(25; 8) = 30 and our construction produces an extremal graph G ∈ EX(25, 8) with D = {45 , 220 }. 2
4
Conclusion
In this paper we have improved the lower bounds for ex(n; t) where n ≤ 49. For n = 23, 24, 25 we established the exact values of ex(n; 8), namely, ex(23; 8) = 28, ex(24; 8) = 29 and ex(25; 8) = 30.
References [1] Abajo, E., C. Balbuena, and A. Di´anez, New families of graphs without short cycles and large size, Discrete Applied Math. 158 (2010), 1127–1135. [2] Abajo, E., and A. Di´anez, Size of Graphs with High Girth, Electronic Notes in Discrete Math. 29 (2007), 179–183. [3] Abajo, E., and A. Di´anez, Exact values of ex(v; {C3 , C4 , . . . , Cn }), Discrete Applied Math. 158 (2010), 1869–1878. [4] Erd˝os, P., Extremal problems in graph theory, Theory Graphs Appl., Proc. Symp. Smolenice 1963, (1964), 29–36. [5] Erd˝os, P., Some recent progress on the extremal problems in graph theory, Congr. Numer. 14 (1975), 3–14. [6] Garnick, D. K., Y. H. H. Kwong, and F. Lazebnik, Extremal graphs without three-cycles or four-cycles, J. Graph Theory 17 (1993), 633–645. [7] Lazebnik, F., and P. Wang, On the structure of extremal graphs of high girth, J. Graph Theory 26 (1997), 147–153.
Electronic Notes in Discrete Mathematics 38 (2011) 615–620 www.elsevier.com/locate/endm
Extremal Graphs without Cycles of Length 8 or Less Kim Marshall 1 School of El. Eng. and Comp. Sc., University of Newcastle, Australia
Mirka Miller 2,3 School of El. Eng. and Comp. Sc., University of Newcastle, Australia Department of Mathematics, University of West Bohemia, Czech Republic Department of Informatics, King’s College London, UK
Joe Ryan 4 School of El. Eng. and Comp. Sc., University of Newcastle, Australia
Abstract Let ex(n; t) denote the maximum number of edges in a graph G having order n without cycles of length t or less. We prove ex(23; 8) = 28, ex(24; 8) = 20 and ex(25; 8) = 30. Furthermore, we present new lower and upper bounds for n ≤ 49 and the extremal numbers when known. Keywords: Extremal graph, extremal number, girth.
1
Email: [email protected] Email: [email protected] 3 This research was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme 4 Email: [email protected] 2
1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2011.10.003
616
1
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
Introduction
Let G be a finite, simple graph of order n = |V (G)|. The extremal number, denoted ex(n; t), is the maximum size m = |E(G)| of a graph having order n and containing no cycle Ck such that k ≤ t. We say that G is an extremal graph, G ∈ EX(n; t), if the number of edges in G is equal to ex(n; t). When the exact value of ex(n; t) is not yet known we use the notation exl (n; t) and exu (n; t) to indicate the lower and upper bounds for the value of ex(n; t).
2
Known Results
In 1940 Tur´an [4] asked: “How many edges must a graph contain that it should certainly have subgraphs of a prescribed structure?”. A class of problems, called Tur´an type problems, stem from this question. The problem of finding the value of the extremal number ex(n; t) belongs to this class of problems. 2 Mantel’s theorem determined ex(n; 3) = n4 . Graphs that attain this bound are the complete bipartite graphs K n2 n2 . In 1975 Erd˝os [5] posed the problem of finding ex(n; 4), that is, the extremal number for graphs that do not contain three-cycles or four-cycles. Lazebnik and Wang [7] along with Garnick et al. [6] provided a summary of results for this problem and constructive lower bounds of ex(n; 4) for n ≤ 200 and exact values for n ≤ 30 and n = 50. More recent work has focused on finding ex(n; t) for t ≥ 4. Abajo and Di´anez [2] found ex(n; t) for all n ≤ (16t − 15)/5. These results include the current known exact values and upper and lower bounds for ex(n; 7) and ex(n; 8) which are displayed in Table 1 and in italics within Table 2. Abajo and Di´anez [3] also obtained bounds and in some cases exact values of ex(n; t) for t ∈ {5, 6, 7}. Improved lower bounds on the extremal number were determined in [1] by the construction of dense graphs for t ∈ {4, 6, 7, 10, 11}. In general the problem of determining the extremal number and constructing extremal graphs is very hard. As a consequence, research activities focus on constructing denser graphs to increase the lower bounds of ex(n; t) and on non-existence proofs to decrease the upper bound.
3
New Results
We begin with a result concerning the lower bound for an infinite series of graphs on n = k 2 vertices having girth 9.
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
n
0
1
2
3
4
5
6
7
8
9
0
0
0
1
2
3
4
5
6
8
9
10
10
12
13
14
16
18
19
20
22
24
20
25
27
29
30
32
34
36
38
40
42
30
45
46
47
49
51
53
55
617
Table 1 Known values of ex(n; 7) for n ≤ 19 by [3].
Theorem 3.1 For k ≥ 3, exl (k 2 ; 8) = 32 k(k − 1). Proof. The proof is by construction. The construction is the subdivision of the complete graph Kk . These graphs include; the cycle C9 for k = 3 and the graphs drawn below for k ∈ {4, 5}. 2 These graphs are known to be extremal for k ≤ 4 (see Table 2). We would like to prove that our construction produces an extremal graph G ∈ EX(n; 8) for k = 5. In order to do this we require Proposition 3.2.
s s @ s s @ s s
s s
s s
s s
@ @s s @ s @s
Subdivision of K4
s sZ s sBBsZ Z s B Zs s s s Zs B s s BZ sZZ B s B Z B B Z B B s s s Z s Bs s Z BBs s sZBs
Subdivision of K5
Proposition 3.2(i) The function ex(n; t) is strictly increasing in n and nondecreasing in t, that is, ex(n; t) ≥ ex(n−1; t)+1 and ex(n; t) ≤ ex(n; t−1). (ii) Let d be the average degree then δ ≤ d ≤ d ≤ d ≤ Δ. (iii) Let G ∈ EX(n; t) and v ∈ G be such that deg(v) = δ. Let G = G − {v}. Then G has order n − 1, g ≥ t and we obtain ex(n; t) − δ ≤ ex(n − 1; t).
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
618
(iv) Combining (i) and (ii) gives the following inequality. ex(n; t) − ex(n − 1; t) ≤ δ ≤
ex(n; t) ∗ 2 ex(n; t) ∗ 2 ≤ ≤Δ n n
We denote by TΔ,δ a tree of height 4 and root r such that deg(r) = Δ and every non-leaf vertex v ∈ V (TΔ,δ ) − {r} has deg(v) = δ. Note that all G ∈ EX(n; 8) contain TΔ,δ as a subgraph. Furthermore, since g ≥ 9 it follows that the vertices in TΔ,δ must all be distinct and edges exist only between parent and child vertices. Note that |V (TΔ,δ )| = 1+Δ+Δ(δ−1)+Δ(δ−1)2 +Δ(δ−1)3 and |E(TΔ,δ )| = Δ + Δ(δ − 1) + Δ(δ − 1)2 + Δ(δ − 1)3 . Let X = V (G − TΔ,δ ). Our results are summarised in Table 2. Previously known values of ex(n; 8) determined by [3] are shown in italics. The new results for the exact value of ex(n; t) are shown in bold text. New lower and upper bounds are given for 26 ≤ n ≤ 49. The table includes only values for n ≤ 49 but our construction can be used to determine lower bounds also for larger n. n
0
1
2
3
4
5
6
7
8
9
0
0
0
1
2
3
4
5
6
7
9
10
10
11
12
14
15
16
18
19
21
22
20
23
25
27
28
29
30
31-32
32-34
33-36
34-38
30
36-40
37-42
39-44
40-46
42-48
43-50
45-52
46-54
47-56
48-58
40
50-60
52-62
54-64
56-66
58-68
60-70
62-73
64-76
66-79
68-82
Table 2 New values of ex(n; 8), for n = 23, 24, 25, and exl (n; 8) and exu (n; 8), for n ≤ 49.
Theorem 3.3 Let G ∈ EX(23; 8). Then |E(G)| = ex(23; 8) = 28. Sketch of Proof. By adding one edge and one vertex to a graph from EX(22; 8) we get the constructive lower bound exl (23; 8) = ex(22; 8)+1 = 28. The upper bound is obtained by combining Proposition 3.2 (i) and results in Table 1, namely, exu (23; 8) = ex(23; 7) = 30. Therefore, 28 ≤ ex(23; 8) ≤ 30. The only values in this range that satisfy the following inequality from Proposition 3.2 (iv) are 28 and 29. ex(23; 8) − ex(22; 8) ≤ δ ≤
ex(23; 8) ∗ 2 ex(23; 8) ∗ 2 ≤ ≤Δ 23 23
K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
619
Therefore, δ = 2 and Δ ≥ 3. Assume ex(23, 8) = 29. We know that Δ ≤ 6 since otherwise |V (T6,2 )| ≥ 25. Therefore, assume Δ = 5 then |V (T5,2 )| ≥ 21 which is okay. However, ex(23; 8) − ex(21, 8) = 29 − 25 = 4 means that removing two vertices from G must remove at least 4 edges of G. That is, in G no two vertices having degree 2 are neighbours. The minimum spanning tree for a graph with Δ ≥ 4, and the additional constraint of no two vertices of degree 2 being neighbours, has at least 25 vertices. Therefore, Δ = 3, and D = {312 , 211 }. However, constructing a tree with root r such that deg(r) = 3 or deg(r) = 2 and no two vertices having degree two are neighbours is not possible. Therefore, ex(23; 8) = 28. 2 Theorem 3.4 Let G ∈ EX(24; 8). Then |E(G)| = ex(24; 8) = 29. Sketch of Proof. By construction, exl (24; 8) = ex(23; 8)+1 = 29. The upper bound is obtained from considering exu (24; 8) = ex(24; 7) = 32. Therefore, 29 ≤ ex(24; 8) ≤ 32. The only values in this range that satisfy Proposition 3.2 (iv) are 29 and 30. Assume that ex(24; 8) = 30. Since ex(24; 8) − ex(21 : 8) = 30 − 25 = 5, there is no path P3 such that all vertices in the path have degree 2. That is, removing 3 vertices must remove at least 5 edges. The minimum spanning tree T5,2 with this additional constraint has 26 vertices. Therefore Δ ≤ 4. We assume Δ = 4 and consider the minimum spanning tree T4,2 of depth 4 which, with the addition forbidden path constraint has |V (T )| = 21, |E(T )| = 20 and degree sequence D = {41 , 34 , 28 , 18 }. From this basis we show that it is impossible to add 3 extra vertices and 10 edges without constructing a cycle of length 8 or less. This gives Δ ≤ 3 and degree sequence D = {312 , 212 }. Considering the twelve vertices of degree 3 as S4 stars, we now resolve the structure of the graph and the lengths of the cycles on which the degree 3 vertices sit. Within this structure, it is not possible to accommodate the required number of edges without constructing cycles smaller than 9. Therefore ex(24; 8) = 29. 2 Theorem 3.5 Let G ∈ EX(25; 8). Then |E(G)| = ex(25; 8) = 30. Sketch of Proof. By construction, exl (25; 8) = ex(24; 8)+1 = 30. The upper bound is obtained from considering exu (25; 8) = ex(25; 7) = 34. Therefore, 30 ≤ ex(25; 8) ≤ 34. The only values in this range that also satisfy Proposition 3.2 (iv) are 30 and 31. Assume that ex(25; 8) = 31. Since ex(25; 8) − ex(21; 8) = 31 − 25 = 6 we cannot have a path of length 3 in G where every vertex has degree 2. That
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K. Marshall et al. / Electronic Notes in Discrete Mathematics 38 (2011) 615–620
is, removing four vertices must remove at least 6 edges. As a consequence of structural properties created by this forbidden path and the number of vertices in |V (TΔ,δ )|, we find that Δ must be less than 4 to avoid small cycles in G. By similar reasoning to Theorem 3.4 we find that Δ = 3 and D = {312 , 213 }. We reason as in Theorem 3.4 by considering the structure forced on the graph by the constraints and show that there must exist a cycle of length less than 9. Therefore ex(25; 8) = 30 and our construction produces an extremal graph G ∈ EX(25, 8) with D = {45 , 220 }. 2
4
Conclusion
In this paper we have improved the lower bounds for ex(n; t) where n ≤ 49. For n = 23, 24, 25 we established the exact values of ex(n; 8), namely, ex(23; 8) = 28, ex(24; 8) = 29 and ex(25; 8) = 30.
References [1] Abajo, E., C. Balbuena, and A. Di´anez, New families of graphs without short cycles and large size, Discrete Applied Math. 158 (2010), 1127–1135. [2] Abajo, E., and A. Di´anez, Size of Graphs with High Girth, Electronic Notes in Discrete Math. 29 (2007), 179–183. [3] Abajo, E., and A. Di´anez, Exact values of ex(v; {C3 , C4 , . . . , Cn }), Discrete Applied Math. 158 (2010), 1869–1878. [4] Erd˝os, P., Extremal problems in graph theory, Theory Graphs Appl., Proc. Symp. Smolenice 1963, (1964), 29–36. [5] Erd˝os, P., Some recent progress on the extremal problems in graph theory, Congr. Numer. 14 (1975), 3–14. [6] Garnick, D. K., Y. H. H. Kwong, and F. Lazebnik, Extremal graphs without three-cycles or four-cycles, J. Graph Theory 17 (1993), 633–645. [7] Lazebnik, F., and P. Wang, On the structure of extremal graphs of high girth, J. Graph Theory 26 (1997), 147–153.