- Email: [email protected]
Deleting edges from Ramsey graphs
Discrete Mathematics 343 (2020) 111743
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Deleting edges from Ramsey graphs✩ ∗
Ye Wang a , , Yusheng Li b a b
School of Stats. & Math., Shanghai Lixin University of Accounting & Finance, Shanghai 201209, China School of Math. Sciences, Tongji University, Shanghai 200092, China
article
info
Article history: Received 19 October 2018 Received in revised form 9 July 2019 Accepted 11 November 2019 Available online xxxx Keywords: Critical Ramsey number Ramsey goodness Complete graph
a b s t r a c t For graphs F , G and H, let F → (G, H) signify that any red–blue edge coloring of F contains either a red G or a blue H. Define the Ramsey number R(G, H) to be the smallest r such that Kr → (G, H), and the complete-critical Ramsey number RK (G, H) to be the largest p such that Kr \ Kp → (G, H), where r = R(G, H). In this note, we shall show a general upper bound for RK (G, H) and determine the exact values of RK (K1,n , Km ), RK (Pn , C4 ) and RK (Fn , K3 ), respectively. © 2019 Elsevier B.V. All rights reserved.
1. Introduction For definitions and notations related to graphs, we refer the reader to West [17]. For vertex disjoint graphs G and H, let G ∪ H be the disjoint union of G and H and G + H the join of G and H obtained by adding additional edges connecting V (G) and V (H) completely. If V (G) ⊆ V (Kn ), we denote by Kn \ G the graph obtained from Kn by deleting the edges of G from Kn . For S ⊆ V (G), we denote by G \ S the subgraph of G induced by V (G) \ S. To distinguish notation Kn \ G and G \ S, we note that G is a graph and S is a subset of vertices, respectively. For graph G, let δ (G) be the minimum degree of G, χ (G) the chromatic number of G, α (G) the independence number of G, and s(G) the size of the smallest color class in a χ (G)-coloring of vertices of G, respectively. A connected graph G is defined [2] to be H-good if |V (G)| ≥ s(H) and R(G, H) = (χ (H) − 1)(|V (G)| − 1) + s(H). For graphs F , G and H, let F → (G, H) signify that any red–blue edge coloring of F contains either a red G or a blue H. Definition 1. Let F , G and H be graphs. We call F a Ramsey graph for G and H if F → (G, H). It is well known that for any given graphs G and H, if n is big enough, then Kn → (G, H), and the Ramsey number is defined as R(G, H) = min{n | Kn → (G, H)}. For r = R(G, H), as Kr is a Ramsey graph with the smallest order r, a problem is to consider the largest graph F such that Kr \ F is still a Ramsey graph. If ‘‘the largest graph’’ here means the largest size of graph, this problem is associated with the size Ramsey number, which was introduced by Erdős, Faudree, Rousseau and Schelp [8] as rˆ (G, H) = min{|E(F )| : F → (G, H)}, where |E(F )| is the number of edges of graph F . ✩ Supported in part by NSFC (11871377, 11701372, 11801371, 11931002) and Shanghai Sailing Program (19YF1435500). ∗ Corresponding author. E-mail addresses: [email protected] (Y. Wang), [email protected] (Y. Li). https://doi.org/10.1016/j.disc.2019.111743 0012-365X/© 2019 Elsevier B.V. All rights reserved.
2
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
In this note, we consider the largest graph F in a class such that Kr \ F → (G, H). Particularly, if the class consists of stars, then we have the following definition by Hook and Isaak. Definition 2 ([12]). For graphs G and H with r = R(G, H), the star-critical Ramsey number is r∗ (G, H) = min n | Kr \ K1,r −n−1 → (G, H) .
{
}
For more results on star-critical Ramsey numbers, see [10–13,18,19], among which the results in [11] are asymptotical formulas. For the graph F being matchings, paths and complete graphs, we shall introduce matching-critical Ramsey number, path-critical Ramsey number and complete-critical Ramsey number in a similar way as follows. Let Sn = K1,n and Mn = nK2 be star and matching of n edges, respectively. Let Pn and Kn be path and complete graph of n vertices, respectively. Set
S = {S1 , S2 , . . . }, M = {M1 , M2 , . . . }, P = {P2 , P3 , . . . }, K = {K2 , K3 , . . . }. Here the graphs are indexed starting at k ≥ 1, not necessarily starting at 1, depending on the graphs to coincide with common notation in graph theory. The indices of stars and matchings start at k = 1, and the indices of paths and complete graphs start at k = 2. Definition 3. Let G = {Gk , Gk+1 , . . .} be a class of graphs, where the fixed integer k ≥ 1 and δ (Gn ) ≥ 1 for n ≥ k. For graphs G and H with r = R(G, H), define the critical Ramsey number RG (G, H) with respect to G as max n | Kr \ Gn → (G, H), Gn ∈ G, |V (Gn )| ≤ r .
{
}
We call RS (G, H) the star-critical Ramsey number. Similarly, we call RM (G, H), RP (G, H) and RK (G, H) the matchingcritical Ramsey number, the path-critical Ramsey number and the complete-critical Ramsey number, respectively. Write RG (G, H) = 0 if there is no Gn ∈ G such that Kr \ Gn → (G, H). Most problems on size Ramsey numbers are far from being solved. This note finds a new way to study this topic. We turn to the notion of critical Ramsey number RG (G, H), which includes the star-critical Ramsey number, the complete-critical Ramsey number, etc. Note that the star-critical Ramsey number RS (G, H) in Definition 3 is related to r∗ (G, H) in Definition 2 as RS (G, H) = r − 1 − r∗ (G, H). Related to the definitions for critical and size Ramsey number, we can define the size critical Ramsey number r˜ (G, H) = max |E(F )| : Kr \ F → (G, H) ,
{
}
and the restricted size Ramsey number r¯ (G, H) =
( ) r
2
− r˜ (G, H) = min{|E(F )| : |V (F )| = r and F → (G, H)},
in which the order of F is restricted as r. Zhang, Broersma and Chen [19] called (G, H) a Ramsey-full pair of graphs if Kr → (G, H), but Kr \ K2 ̸ → (G, H). Thus (r )if (G, H) is the known Ramsey-full pair of graphs (Km , Kn ), (Kn , mK2 ), (nK3 , mK3 ) or (nK4 , mK3 ) in [19], we have r¯ (G, H) = 2 . More results on the (restricted) size Ramsey number of graphs can be found in [1,9]. It is clear that for graphs G and H without isolated vertices, rˆ (G, H) ≤ r¯ (G, H).
(1)
It is interesting to answer when rˆ (G, H) = r¯ (G, H). From the definitions, rˆ (G, H) ≤ r¯ (G, H) ≤
. If rˆ (G, H) =
(r )
(r )
2
2
, we
have rˆ (G, H) = r¯ (G, H).(As observed ) (rby ) Chvátal [8], equality holds in (1) when G and H are both complete graphs. Erdős [7] n+2m−2 showed rˆ (mK2 , Kn ) = = , and then the equality also holds for matching and complete graph. However, the 2 2 inequality (1) is not an equality generally. For example, if m and n are both even, it is shown that (K1,m , K1,n ) is Ramsey-full in [16]. Since r(K1,m , K1,n ) = m + n − 1, we have rˆ (K1,m , K1,n ) = m + n − 1 < r¯ (K1,m , K1,n ) =
(
m+n−1 2
) ,
where m and n are both even and m + n > 4. Let us focus on the complete-critical Ramsey number RK (G, H). Clearly RK (G, H) = 0 for Ramsey-full pair of graphs (G, H). It is shown in [16] that RK (Sm , Sn ) = 0 if m and n are both even or m = n = 1, and RK (Sm , Sn ) = m + n − 1 otherwise. And RK (Mm , Mn ) = n if n ≥ m ≥ 1 and n ≥ 2. In this note, we shall give a general upper bound for RK (G, H), which is sharp for cases in Theorems 1 and 3.
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
Theorem 1.
For any integers n, m ≥ 2, RK (K1,n , Km ) = n.
Theorem 2.
For any integer n ≥ 4, RK (Pn , C4 ) = ⌈ 2n ⌉.
3
In the following result, Fn = K1 + nK2 is called a fan. Theorem 3.
For any integer n ≥ 2, RK (Fn , K3 ) = n.
2. Preliminaries Lemma 1.
Let G be a connected graph with at least two vertices. Then
RK (G, H) ≤ α (G) + r − (χ (H) − 1)(|V (G)| − 1) − 1,
(2)
where r = R(G, H). Particularly, if G is H-good, then RK (G, H) ≤ α (G) + s(H) − 1.
(3)
Proof. We consider two cases depending on the value of χ (H) to show that RK (G, H) ≤ α (G) + r − (χ (H) − 1)(|V (G)|− 1) − 1. Case 1. χ (H) = 1. In this case, we have H is an empty graph, and thus r = |V (H)|. Clearly RK (G, H) = r = |V (H)| ≤ α (G) + |V (H)| − 1, as desired. Case 2. χ (H) ≥ 2. Consider the graph F = Kr \ Kt , where t = α (G) + r − (χ (H) − 1)(|V (G)| − 1). Note that F = Kr −t + tK1 = K(χ (H)−2)(|V (G)|−1)+|V (G)|−α (G)−1 + tK1 . Color the edges of F as follows, in which R and B are the subgraphs induced by red edges and blue edges, respectively. Partition the vertex set of K(χ (H)−2)(|V (G)|−1)+|V (G)|−α (G)−1 into χ (H) − 1 parts: each of χ (H) − 2 parts contains |V (G)| − 1 vertices, and the last part contains |V (G)| − α (G) − 1 vertices. We then color the edges of F by red and blue such that R = χ (H) − 2 K|V (G)|−1 ∪ K|V (G)|−α (G)−1 + tK1 ,
(
)
(
)
and all the other edges blue. There are two types of components of R: one type is K|V (G)|−1 that contains no G as G is connected, and the other type is K|V (G)|−α (G)−1 + tK1 . The last component of R contains no G as the independence number of any subgraph induced by |V (G)| vertices is greater than α (G). On the other hand, B is a (χ (H) − 1)-partite graph with B = K|V (G)|−1,|V (G)|−1,...,|V (G)|−1,|V (G)|−α (H)+t −1 , then the chromatic number of B is χ (H) − 1. Suppose B contains an H, we have χ (H) ≤ χ (B) = χ (H) − 1, a contradiction. Thus B contains no H, which implies the desired upper bound. If G is H-good, we have that r = (χ (H) − 1)(|V (G)| − 1) + s(H). Then RK (G, H) ≤ α (G) + r − (χ (H) − 1)(|V (G)| − 1) − 1 = α (G) + s(H) − 1. □ We will see that the bound (3) is sharp for many cases when G is H-good. If G is not H-good, the bound (3) does not hold. For example, if one of m and n is odd and n ≥ m ≥ 2, it is shown that R(Sm , Sn ) = m + n in [4,6] and RK (Sm , Sn ) = m + n − 1 in [16]. Thus RK (Sm , Sn ) > α (Sm ) + s(Sn ) − 1 = m. This example tells us that the bound (2) is also sharp for cases that G is not H-good. Lemma 2 ([5]). For any integers m ≥ 2 and n ≥ 1, R(K1,n , Km ) = n(m − 1) + 1. Proof of Theorem 1. The upper bound follows from Lemma 1. For the lower bound, consider the graph G = G1 + G2 with G1 = Kn(m−2)+1 and G2 = nK1 . We will proceed by induction on m. Case 1. m = 2. Then for any n ≥ 2, G = K1,n → (K1,n , K2 ). Case 2. m ≥ 3. Assume the assertion holds for m − 1. For any 2-coloring of the edges of graph G, suppose that G contains no blue Km , and we shall show that G must contain a red K1,n . By induction, G contains a blue Km−1 . If there is a blue Km−1 in G1 , since G contains no blue Km , any vertex in V (G) \ V (Km−1 ) is adjacent to at least one vertex in V (Km−1 ) by a red edge. So |V (G) \ V (Km−1 )| = (n − 1)(m − 1) + 1, then there is a vertex in V (Km−1 ) adjacent to at least n vertices in V (G) \ V (Km−1 ) by red edges, producing a red K1,n . If there is no blue Km−1 in G1 , then the blue Km−1 in G has m − 2 vertices in V (G1 ), and one vertex in V (G2 ), respectively. As there is no blue Km−1 in G1 , by the similar argument, there must be a vertex in Km−2 adjacent to at least n vertices in V (G1 ) \ V (Km−2 ) by red edges, producing a red K1,n . □
4
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
Fig. 1. u1 ∈ V (G1 ), v1 , v2 , u2 ∈ V (G2 ).
3. Paths versus C4 For the figures in this section, red edges are solid and blue edges are dashed. The path Pn and the cycle Cn will be denoted as p1 p2 . . . pn and c1 c2 . . . cn with pi ∈ V (Pn ) and ci ∈ V (Cn ) for 1 ≤ i ≤ n, respectively. Lemma 3 ([3]). For all n ≥ 3, R(Pn , C4 ) = n + 1. Lemma 4 ([12]). For all n ≥ 3, RS (Pn , C4 ) = n − 3. Proof of Theorem 2. For the upper bound, color the edges of K⌊n/2⌋ + (⌈n/2⌉ + 1)K1 red and blue such that the red graph is isomorphic to K⌊n/2⌋−1 + (⌈n/2⌉ + 1)K1 and the blue graph is isomorphic to K1,n . So it contains neither a red path Pn nor a blue cycle C4 , which yields RK (Pn , C4 ) ≤ ⌈n/2⌉. For the lower bound, we will see K⌊n/2⌋+1 + ⌈n/2⌉K1 → (Pn , C4 ) by induction on n. For n = 4, Lemmas 3 and 4 imply K3 + 2K1 → (P4 , C4 ). For n ≥ 5, we consider the parity of n. Case 1. For n even, consider the graph G = G1 + G2 with G1 = Kn/2+1 and G2 = (n/2)K1 . For any vertex u1 ∈ V (G1 ), by induction, we obtain that G \ {u1 } contains a red path Pn−1 , denoted by P with endpoints v1 and v2 . Let u2 be the vertex of V (G) \ (V (P) ∪ {u1 }). Suppose that G contains no red Pn . Subcase 1.1. Let v1 , v2 ∈ V (G1 ). Since G contains no red Pn , the edges u1 v1 , u1 v2 , u2 v1 and u2 v2 are all blue, producing a blue C4 . Subcase 1.2. Let v1 ∈ V (G1 ), v2 ∈ V (G2 ). Then u2 ∈ V (G2 ). Since G contains no red Pn , the edges u1 v1 , u1 v2 and u2 v1 are all blue. Then v1 v2 must be blue, otherwise u1 and u2 would be adjacent to all the vertices in V (P) ∩ V (G1 ) by blue edges, which forms a blue C4 . If u1 u2 is blue, there is a blue C4 : u1 u2 v1 v2 . Assume u1 u2 is red. Let u3 be the vertex adjacent to v2 in P. Since u3 ∈ V (G1 ) and u1 u2 is red, we obtain u1 u3 and u2 u3 are both blue, producing a blue C4 : u1 u3 u2 v1 . Subcase 1.3. Let v1 , v2 ∈ V (G2 ). If u2 ∈ V (G1 ), G contains either a red Pn or a blue C4 , similar to Subcase 1.1. Thus we suppose that u2 ∈ V (G2 ). Since there is at least one vertex of V (G1 ) between any two vertices of V (G2 ) in the path P, by the Pigeonhole Principle, there are three vertices of V (G1 ), denoted by p1 , p2 , p3 , such that p1 p2 p3 ⊆ P, or there are four vertices of V (G1 ), denoted by p′1 , p′2 , p′3 , p′4 , such that p′1 p′2 , p′3 p′4 ⊆ P. If p1 p2 p3 ⊆ P, then v1 p2 and v2 p2 are both blue, otherwise we obtain a new red path on n − 1 vertices with one endpoint in V (G1 ) and the other endpoint in V (G2 ), and it becomes Subcase 1.2. Thus, u1 v1 p2 v2 forms a blue cycle C4 . If p′1 p′2 , p′3 p′4 ⊆ P, v2 p′3 must be blue (see Fig. 1), otherwise we obtain a new red path on n − 1 vertices with endpoints p′4 and v1 . Then v1 p′3 is red, or we can get a blue cycle u1 v1 p′3 v2 . Similarly, v2 p′2 is also red, and thus we get a new red path on n − 1 vertices with endpoints p′1 and p′4 , which becomes Subcase 1.1. ( ) Case 2. For n odd, consider the graph G = G1 + G2 with G1 = K(n+1)/2 and G2 = (n + 1)/2 K1 . For any vertex u1 ∈ V (G2 ), by induction, we obtain that G \ {u1 } contains a red path Pn−1 , denoted by P with endpoints v1 and v2 . Let u2 be the vertex of V (G) \ (V (P) ∪ {u1 }). Suppose that G contains no red Pn . Subcase 2.1. Let v1 , v2 ∈ V (G1 ). Since G contains no red Pn , the edges u1 v1 , u1 v2 , u2 v1 and u2 v2 are all blue, producing a blue C4 . Subcase 2.2. Let v1 ∈ V (G1 ), v2 ∈ V (G2 ). If u2 ∈ V (G1 ), it is similar to Subcase 1.2. Now suppose that u2 ∈ V (G2 ). By the Pigeonhole Principle, there are two vertices of V (G1 ), other than v1 , denoted by p1 , p2 , such that p1 p2 ⊆ P (see Fig. 2). Since G contains no red Pn , the edges u1 v1 , u2 v1 and v1 v2 are blue. And p1 v2 is also blue, otherwise we get a new red path on n − 1 vertices with endpoints p2 and v1 , resulting in Subcase 2.1. If p1 u2 is blue, then we have a blue C4 : u2 v1 v2 p1 . So we suppose that p1 u2 is red, and using the same method, p1 u1 is also red. Then u1 p2 and u2 p2 are both blue, or we can obtain a red path on n vertices by replacing p1 p2 with p1 u1 p2 and p1 u2 p2 , respectively. Thus, we obtain a blue C4 : v1 u1 p2 u2 . Subcase 2.3. Let v1 , v2 ∈ V (G2 ). If u2 ∈ V (G1 ), by the Pigeonhole Principle, there are two vertices of V (G1 ), denoted by p1 , p2 , such that p1 p2 ⊆ P (see Fig. 3). Thus v1 p2 and v2 p1 are blue, or we can obtain a red path on n − 1 vertices with one endpoint in V (G1 ) and the other endpoint in V (G2 ). Since G contains no red Pn , at least one of u1 p1 and u1 p2 is blue, say u1 p2 . Then u1 u2 is red, or we get a blue C4 : v1 u2 u1 p2 . Denote the vertices adjacent to v1 and v2 in path P by p and p′ , and then we get a blue C4 : u1 pu2 p′ .
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
5
Fig. 2. v1 ∈ V (G1 ), v2 , u1 , u2 ∈ V (G2 ).
Fig. 3. u2 ∈ V (G1 ), v1 , v2 , u1 ∈ V (G2 ).
Fig. 4. v1 , v2 , u1 , u2 ∈ V (G2 ).
If u2 ∈ V (G2 ), there are either three vertices of V (G1 ), denoted by p1 , p2 , p3 , such that p1 p2 p3 ⊆ P, or three pairs of vertices of V (G1 ), denoted by p′1 , p′2 , p′3 , p′4 , and p′5 , p′6 , such that p′1 p′2 , p′3 p′4 , p′5 p′6 ⊆ P. If p1 p2 p3 ⊆ P (see Fig. 4), then v1 p2 , v1 p3 , v2 p1 , v2 p2 are all blue. If p2 u1 is blue, then p1 u1 and p3 u1 are both red, or we get a blue C4 . And now we have a new red path on n − 1 vertices by replacing p2 with u1 , which becomes the case where v1 , v2 ∈ V (G2 ) and u2 ∈ V (G1 ). The case will be similar if p2 u2 is blue. Thus, we suppose that p2 u1 and p2 u2 are both red, and then p1 u1 , p3 u1 , p1 u2 , p3 u2 are all blue, producing a blue cycle C4 . If p′1 p′2 , p′3 p′4 , p′5 p′6 ⊆ P (see Fig. 4), v1 p′i and v2 p′j are both blue, for i ∈ {2, 4, 6}, j ∈ {1, 3, 5}, and at least one of p′3 u1 and p′4 u1 is blue. Without loss of generality, suppose that p′3 u1 is blue. Since there is no blue C4 , p′1 u1 and p′5 u1 are red, and p′2 u1 and p′6 u1 are blue, along with p′2 v1 , p′6 v1 , producing a blue C4 . □ 4. Fans versus K3 Before proceeding to the proof, we need some notation and lemmas. If there is a red–blue edge coloring of F such that there is neither a red G nor a blue H, we call such a coloring to be a (G, H)-free coloring and the graph F is called a (G, H)-free graph. In a red–blue edge coloring of F , the subgraph induced by the red edges is denoted by F R and the subgraph induced by the blue edges is denoted by F B . And for a vertex x of F , we denote the numbers of the red and blue neighbors of x in F by degFR (x) and degFB (x), respectively. Lemma 5 ([14]). For all n ≥ 2, R(Fn , K3 ) = 4n + 1. Lemma 6 ([13]). For all n ≥ 2, RS (Fn , K3 ) = 2n − 2. Lemma 7 ([13]). Let n ≥ 2 be an integer and r = R(Fn , K3 ) = 4n + 1. Define a class of red–blue edge colored K4n as H = {H1 , H2 , . . . , H2n }, such that Hi : HiB = K2n,2n \ iK2 HiR = K4n \ HiB , where iK2 is i independent edges of K2n,2n . For any (Fn , K3 )-free coloring of Kr −1 , the resulting graph must belong to the class H.
6
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
Lemma 8 ([15]). For all n ≥ 2, R(nK2 , K3 ) = 2n + 1. Lemma 9 ([13]). Let n ≥ 2 be an integer and r = R(nK2 , K3 ) = 2n + 1. Define a class of red–blue edge colored K2n as H = {H1 , H2 , . . . , H⌈n/2⌉−1 }, such that Hi : HiR = K2i+1 ∪ K2n−2i−1 HiB = K2i+1,2n−2i−1 , where 1 ≤ i ≤ ⌈n/2⌉ − 1. For any (nK2 , K3 )-free coloring of Kr −1 , the resulting graph must belong to the class H. Proof of Theorem 3. The upper bound follows from Lemma 1. For the lower bound, consider the graph G = G1 + G2 with G1 = K3n+1 and G2 = nK1 . We will show that G contains either a red Fn with the central vertex in V (G1 ) or a blue K3 by induction on n. Case 1. Let n = 2. By Lemma 6, we obtain K9 \ K2 → (F2 , K3 ). If G contains no blue K3 , then G must contain a red F2 . Assume that there is no red F2 with the central vertex in V (G1 ), and denote the set of the central vertices of all red F2 in G by V with V ⊆ V (G2 ). Let V (G2 ) = {v1 , v1′ } with v1 ̸ = v1′ . Subcase 1.1. Let V = {v1 } (or V = {v1′ }). Since G \ {v1 } ̸ → (F2 , K3 ), by Lemma 7, G \ {v1 } must belong to the defined class H for n = 2. Since v1 is adjacent to all the vertices in V (G1 ), v1 has at most one red neighbor in each red clique, otherwise we obtain a red F2 with central vertex in V (G1 ). Then v1 has at least two blue neighbors in each red clique, producing a blue K3 . Subcase 1.2. Let V = {v1 , v1′ }. Let F and F ′ be two red F2 in G with V (F ) = {v1 , u1 , u2 , u3 , u4 } and V (F ′ ) = {v1′ , u′1 , u′2 , u′3 , u′4 }, respectively, and ui , u′i ∈ V (G1 ), for 1 ≤ i ≤ 4. Then |V (F ) ∩ V (F ′ )| ≥ 1. If |V (F ) ∩ V (F ′ )| = 1 or |V (F ) ∩ V (F ′ )| = 3, there is a red F2 with central vertex in V (G1 ) trivially. If |V (F ) ∩ V (F ′ )| = 2 and |E(F ) ∩ E(F ′ )| = 0, there is also a red F2 with central vertex in V (G1 ) trivially, and then we assume that |V (F ) ∩ V (F ′ )| = 2 and |E(F ) ∩ E(F ′ )| = 1. Without loss of generality, let u1 = u′1 , u2 = u′2 , and u1 u2 , u3 u4 , u′3 u′4 are red edges in G. Since G contains no red F2 with central vertex in V (G1 ), u2 is adjacent to u3 , u′3 , u4 , u′4 by blue edges in G. And then {u3 , u′3 , u4 , u′4 } induces a red clique in G. Along with the red edges v1 u3 and v1 u4 , we obtain a red F2 with central vertex u3 . If |V (F ) ∩ V (F ′ )| = 4 and |E(F ) ∩ E(F ′ )| = 0, there is also a red F2 with central vertex in V (G1 ) trivially, and then we assume that |V (F ) ∩ V (F ′ )| = 4 and |E(F ) ∩ E(F ′ )| = 2. Without loss of generality, let u1 = u′1 , u2 = u′2 , u3 = u′3 , u4 = u′4 , and u1 u2 , u3 u4 are red edges in G. For any vertex u ∈ V (G1 ) \ {u1 , u2 , u3 , u4 }, there is at least one blue edge in {uu1 , uu2 , uu3 , uu4 }. If uu1 or uu2 is blue, since u1 u3 , u1 u4 , u2 u3 , u2 u4 are all blue, then uu3 and uu4 are both red. Similarly, if uu3 or uu4 is blue, uu1 and uu2 are both red. Since there are three vertices in V (G1 ) \{u1 , u2 , u3 , u4 }, there are at least two vertices, denoted by u and u′ , such that uu1 , uu2 , u′ u1 , u′ u2 or uu3 , uu4 , u′ u3 , u′ u4 are all red. Without loss of generality, we assume that uu1 , uu2 , u′ u1 , u′ u2 are all red. Since G contains no red F2 with central vertex in V (G1 ), uv1 and u′ v1 are both blue. Then uu′ must be red, and {u, u′ , u1 , u2 } induces a red clique in G, along with v1 u1 and v1 u2 , forming a red F2 with central vertex u1 . Thus all the cases produce a red F2 with central vertex in V (G1 ), which contradicts our assumption. Case 2. Let n ≥ 3. Assume the assertion holds for n − 1. If the edges between V (G1 ) and V (G2 ) in G are completely red, then for any vertex x ∈ V (G1 ), degGR (x) ≤ n − 1, 1 otherwise we obtain a red Fn with central vertex x. Then degGB (x) ≥ 2n + 1. Suppose that G contains no blue K3 , and then 1 the blue neighbors of x induce a red clique containing Fn in G1 , a contradiction. Thus there exists at least one blue edge between V (G1 ) and V (G2 ), denoted by the edge u1 v with u1 ∈ V (G1 ) and v ∈ V (G2 ). If degGB (u1 ) = 0, by Lemma 8, G1 contains a red Fn with central vertex u1 or a blue K3 . Then degGB (u1 ) ≥ 1, and we 1 1 denote u2 the vertex adjacent to u1 by a blue edge in G1 . Similarly, we can also find vertex u3 ∈ V (G1 ) \ {u1 , u2 } such that u2 u3 is blue. Since G contains no blue K3 , v u2 and u1 u3 are red. By induction, G \ {v, u1 , u2 , u3 } must contain a red Fn−1 with central vertex u ∈ V (G1 ). Subcase 2.1. Assume uv is red. Since v u2 is red, uu2 must be blue. As u2 u1 and u2 u3 are blue, we have uu1 and uu3 are both red. Thus, V (Fn−1 ) ∪ {u1 , u3 } produces a red Fn in G, a contradiction. Subcase 2.2. Assume uv is blue. Then uu1 is red. Since u1 u3 is red, uu3 is blue. Then v u3 and uu2 are red. Note that u is adjacent to any vertex in V (G) \ (V (Fn−1 ) ∪ {u1 , u2 }) by a blue edge. Otherwise, we obtain either a red Fn with central vertex u or a blue K3 in G. Thus, vertex u has 2n blue neighbors and also has 2n red neighbors. And these blue neighbors induce a completely red graph in G, denoted by Q . If |V (Q ) ∩ V (G1 )| > n, for vertex u′ ∈ V (Q ) ∩ V (G1 ), at least one of u1 u′ and u2 u′ is red. Since |V (Q ) ∩ V (G1 )| ≥ n ≥ 3, one of the vertices u1 and u2 , say u1 , has at least two red neighbors in V (Q ) ∩ V (G1 ). Then we obtain a red Fn with V (Fn ) = {u1 } ∪ V (Q ) and central vertex in V (G1 ). If V (Q ) ∩ V (G1 ) = n, then the blue neighbors of vertex u induce graph Q = Kn + nK1 = Q1 + Q2 in G, and the red neighbors of vertex u induce graph Q ′ = K2n in G. Since Q is completely red, and u3 ∈ Q1 , v ∈ Q2 , it is easily shown that u1 is adjacent to every vertex in V (Q1 ) by a red edge, and u2 is adjacent to every vertex in V (Q2 ) by a red edge, otherwise we can get either a red Fn with central vertex in V (Q1 ) or a blue K3 . If degQB ′ (u2 ) ≥ 2, let u1 and u′ be the blue neighbors of u2 in Q ′ . Similar to u1 , u′ must be adjacent to every vertex in V (Q1 ) by a red edge. Since u1 u′ is red, we obtain a red Fn with
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
7
central vertex in V (Q1 ) using vertices V (Q ) ∪ {u1 , u′ } \ {v}. Then we assume that degQB ′ (u2 ) = 1. Since degQR ′ (u2 ) = 2n − 2 and Q ′ is (nK2 , K3 )-free, by Lemma 9, Q ′ is the graph H0 in H with degQB ′ (u1 ) = 2n − 1. And by the similar argument above, for any vertex in V (Q ′ ) \ {u1 }, this vertex is adjacent to every vertex in V (Q2 ) by a red edge. As V (Q ′ ) \ {u1 } induces a red K2n−1 in G, along with the red edges between V (Q ′ ) \ {u1 } and V (Q2 ), we obtain a red Fn with central vertex in V (Q ′ ). □ Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors are grateful to the referees for their invaluable comments. Particularly, following their comments for the proof of Lemma 1 (original Theorem 1), we obtain a new bound (2) such that the original bound (3) is a special case of (2). Also, the paragraph after Definition 3 comes from one of reports. References [1] S.A. Burr, A survey of noncomplete Ramsey theory for graphs, Ann. New York Acad. Sci. 328 (1979) 58–75. [2] S.A. Burr, Ramsey numbers involving graphs with long suspended paths, J. Lond. Math. Soc. 24 (1981) 405–413. [3] S.A. Burr, P. Erdős, R.J. Faudree, C.C. Rousseau, R.H. Schelp, Some complete bipartite graph-tree Ramsey numbers, Ann. Discrete Math. 41 (1989) 79–89. [4] S.A. Burr, J. Roberts, On Ramsey numbers for stars, Util. Math. 4 (1973) 217–220. [5] V. Chvátal, Tree-complete graph Ramsey numbers, J. Graph Theory 1 (1977) 93. [6] V. Chvátal, F. Harary, Generalized Ramsey theory for graphs II, small diagonal numbers, Proc. Amer. Math. Soc. 32 (1972) 389–394. [7] P. Erdős, R.J. Faudree, Size Ramsey numbers involving matchings, Finite Infin. Sets 37 (1984) 247–264. [8] P. Erdős, R.J. Faudree, C. Rousseau, R. Schelp, The size Ramsey numbers, Period. Math. Hungar. 9 (1978) 145–161. [9] R.J. Faudree, R.H. Schelp, A survey of results on the size Ramsey number, in: Paul Erdös and His Mathematics II(Budapest, 1999), in: Bolyai Soc. Math. Stud., vol. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 291–309. [10] S. Haghi, H. Maimani, A. Seify, Star-critical Ramsey numbers of Fn versus K4 , Discrete Appl. Math. 217 (2017) 203–209. [11] Y. Hao, Q. Lin, Star-critical Ramsey numbers for large generalized fans and books, Discrete Math. 341 (2018) 3385–3393. [12] J. Hook, G. Isaak, Star-critical Ramsey numbers, Discrete Appl. Math. 159 (2011) 328–334. [13] Z. Li, Y. Li, Some star-critical Ramsey numbers, Discrete Appl. Math. 181 (2015) 301–305. [14] Y. Li, C. Rousseau, Fan-complete graph Ramsey numbers, J. Graph Theory 23 (1996) 413–420. [15] P. Lorimer, The Ramsey numbers for stripes and one complete graph, J. Graph Theory 8 (1984) 177–184. [16] Y. Wang, Y. Li, Redundant edges in Ramsey graphs, Discrete Appl. Math., revised. [17] D.B. West, Introduction to Graph Theory, second ed., Prentice Hall, New Jersey, 2000. [18] Y. Wu, Y. Sun, S. Radziszowski, Wheel and star-critical Ramsey numbers for quadrilateral, Discrete Appl. Math. 186 (2015) 260–271. [19] Y. Zhang, H. Broersma, Y. Chen, On star-critical and upper size Ramsey numbers, Discrete Appl. Math. 202 (2016) 174–180.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Deleting edges from Ramsey graphs✩ ∗
Ye Wang a , , Yusheng Li b a b
School of Stats. & Math., Shanghai Lixin University of Accounting & Finance, Shanghai 201209, China School of Math. Sciences, Tongji University, Shanghai 200092, China
article
info
Article history: Received 19 October 2018 Received in revised form 9 July 2019 Accepted 11 November 2019 Available online xxxx Keywords: Critical Ramsey number Ramsey goodness Complete graph
a b s t r a c t For graphs F , G and H, let F → (G, H) signify that any red–blue edge coloring of F contains either a red G or a blue H. Define the Ramsey number R(G, H) to be the smallest r such that Kr → (G, H), and the complete-critical Ramsey number RK (G, H) to be the largest p such that Kr \ Kp → (G, H), where r = R(G, H). In this note, we shall show a general upper bound for RK (G, H) and determine the exact values of RK (K1,n , Km ), RK (Pn , C4 ) and RK (Fn , K3 ), respectively. © 2019 Elsevier B.V. All rights reserved.
1. Introduction For definitions and notations related to graphs, we refer the reader to West [17]. For vertex disjoint graphs G and H, let G ∪ H be the disjoint union of G and H and G + H the join of G and H obtained by adding additional edges connecting V (G) and V (H) completely. If V (G) ⊆ V (Kn ), we denote by Kn \ G the graph obtained from Kn by deleting the edges of G from Kn . For S ⊆ V (G), we denote by G \ S the subgraph of G induced by V (G) \ S. To distinguish notation Kn \ G and G \ S, we note that G is a graph and S is a subset of vertices, respectively. For graph G, let δ (G) be the minimum degree of G, χ (G) the chromatic number of G, α (G) the independence number of G, and s(G) the size of the smallest color class in a χ (G)-coloring of vertices of G, respectively. A connected graph G is defined [2] to be H-good if |V (G)| ≥ s(H) and R(G, H) = (χ (H) − 1)(|V (G)| − 1) + s(H). For graphs F , G and H, let F → (G, H) signify that any red–blue edge coloring of F contains either a red G or a blue H. Definition 1. Let F , G and H be graphs. We call F a Ramsey graph for G and H if F → (G, H). It is well known that for any given graphs G and H, if n is big enough, then Kn → (G, H), and the Ramsey number is defined as R(G, H) = min{n | Kn → (G, H)}. For r = R(G, H), as Kr is a Ramsey graph with the smallest order r, a problem is to consider the largest graph F such that Kr \ F is still a Ramsey graph. If ‘‘the largest graph’’ here means the largest size of graph, this problem is associated with the size Ramsey number, which was introduced by Erdős, Faudree, Rousseau and Schelp [8] as rˆ (G, H) = min{|E(F )| : F → (G, H)}, where |E(F )| is the number of edges of graph F . ✩ Supported in part by NSFC (11871377, 11701372, 11801371, 11931002) and Shanghai Sailing Program (19YF1435500). ∗ Corresponding author. E-mail addresses: [email protected] (Y. Wang), [email protected] (Y. Li). https://doi.org/10.1016/j.disc.2019.111743 0012-365X/© 2019 Elsevier B.V. All rights reserved.
2
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
In this note, we consider the largest graph F in a class such that Kr \ F → (G, H). Particularly, if the class consists of stars, then we have the following definition by Hook and Isaak. Definition 2 ([12]). For graphs G and H with r = R(G, H), the star-critical Ramsey number is r∗ (G, H) = min n | Kr \ K1,r −n−1 → (G, H) .
{
}
For more results on star-critical Ramsey numbers, see [10–13,18,19], among which the results in [11] are asymptotical formulas. For the graph F being matchings, paths and complete graphs, we shall introduce matching-critical Ramsey number, path-critical Ramsey number and complete-critical Ramsey number in a similar way as follows. Let Sn = K1,n and Mn = nK2 be star and matching of n edges, respectively. Let Pn and Kn be path and complete graph of n vertices, respectively. Set
S = {S1 , S2 , . . . }, M = {M1 , M2 , . . . }, P = {P2 , P3 , . . . }, K = {K2 , K3 , . . . }. Here the graphs are indexed starting at k ≥ 1, not necessarily starting at 1, depending on the graphs to coincide with common notation in graph theory. The indices of stars and matchings start at k = 1, and the indices of paths and complete graphs start at k = 2. Definition 3. Let G = {Gk , Gk+1 , . . .} be a class of graphs, where the fixed integer k ≥ 1 and δ (Gn ) ≥ 1 for n ≥ k. For graphs G and H with r = R(G, H), define the critical Ramsey number RG (G, H) with respect to G as max n | Kr \ Gn → (G, H), Gn ∈ G, |V (Gn )| ≤ r .
{
}
We call RS (G, H) the star-critical Ramsey number. Similarly, we call RM (G, H), RP (G, H) and RK (G, H) the matchingcritical Ramsey number, the path-critical Ramsey number and the complete-critical Ramsey number, respectively. Write RG (G, H) = 0 if there is no Gn ∈ G such that Kr \ Gn → (G, H). Most problems on size Ramsey numbers are far from being solved. This note finds a new way to study this topic. We turn to the notion of critical Ramsey number RG (G, H), which includes the star-critical Ramsey number, the complete-critical Ramsey number, etc. Note that the star-critical Ramsey number RS (G, H) in Definition 3 is related to r∗ (G, H) in Definition 2 as RS (G, H) = r − 1 − r∗ (G, H). Related to the definitions for critical and size Ramsey number, we can define the size critical Ramsey number r˜ (G, H) = max |E(F )| : Kr \ F → (G, H) ,
{
}
and the restricted size Ramsey number r¯ (G, H) =
( ) r
2
− r˜ (G, H) = min{|E(F )| : |V (F )| = r and F → (G, H)},
in which the order of F is restricted as r. Zhang, Broersma and Chen [19] called (G, H) a Ramsey-full pair of graphs if Kr → (G, H), but Kr \ K2 ̸ → (G, H). Thus (r )if (G, H) is the known Ramsey-full pair of graphs (Km , Kn ), (Kn , mK2 ), (nK3 , mK3 ) or (nK4 , mK3 ) in [19], we have r¯ (G, H) = 2 . More results on the (restricted) size Ramsey number of graphs can be found in [1,9]. It is clear that for graphs G and H without isolated vertices, rˆ (G, H) ≤ r¯ (G, H).
(1)
It is interesting to answer when rˆ (G, H) = r¯ (G, H). From the definitions, rˆ (G, H) ≤ r¯ (G, H) ≤
. If rˆ (G, H) =
(r )
(r )
2
2
, we
have rˆ (G, H) = r¯ (G, H).(As observed ) (rby ) Chvátal [8], equality holds in (1) when G and H are both complete graphs. Erdős [7] n+2m−2 showed rˆ (mK2 , Kn ) = = , and then the equality also holds for matching and complete graph. However, the 2 2 inequality (1) is not an equality generally. For example, if m and n are both even, it is shown that (K1,m , K1,n ) is Ramsey-full in [16]. Since r(K1,m , K1,n ) = m + n − 1, we have rˆ (K1,m , K1,n ) = m + n − 1 < r¯ (K1,m , K1,n ) =
(
m+n−1 2
) ,
where m and n are both even and m + n > 4. Let us focus on the complete-critical Ramsey number RK (G, H). Clearly RK (G, H) = 0 for Ramsey-full pair of graphs (G, H). It is shown in [16] that RK (Sm , Sn ) = 0 if m and n are both even or m = n = 1, and RK (Sm , Sn ) = m + n − 1 otherwise. And RK (Mm , Mn ) = n if n ≥ m ≥ 1 and n ≥ 2. In this note, we shall give a general upper bound for RK (G, H), which is sharp for cases in Theorems 1 and 3.
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
Theorem 1.
For any integers n, m ≥ 2, RK (K1,n , Km ) = n.
Theorem 2.
For any integer n ≥ 4, RK (Pn , C4 ) = ⌈ 2n ⌉.
3
In the following result, Fn = K1 + nK2 is called a fan. Theorem 3.
For any integer n ≥ 2, RK (Fn , K3 ) = n.
2. Preliminaries Lemma 1.
Let G be a connected graph with at least two vertices. Then
RK (G, H) ≤ α (G) + r − (χ (H) − 1)(|V (G)| − 1) − 1,
(2)
where r = R(G, H). Particularly, if G is H-good, then RK (G, H) ≤ α (G) + s(H) − 1.
(3)
Proof. We consider two cases depending on the value of χ (H) to show that RK (G, H) ≤ α (G) + r − (χ (H) − 1)(|V (G)|− 1) − 1. Case 1. χ (H) = 1. In this case, we have H is an empty graph, and thus r = |V (H)|. Clearly RK (G, H) = r = |V (H)| ≤ α (G) + |V (H)| − 1, as desired. Case 2. χ (H) ≥ 2. Consider the graph F = Kr \ Kt , where t = α (G) + r − (χ (H) − 1)(|V (G)| − 1). Note that F = Kr −t + tK1 = K(χ (H)−2)(|V (G)|−1)+|V (G)|−α (G)−1 + tK1 . Color the edges of F as follows, in which R and B are the subgraphs induced by red edges and blue edges, respectively. Partition the vertex set of K(χ (H)−2)(|V (G)|−1)+|V (G)|−α (G)−1 into χ (H) − 1 parts: each of χ (H) − 2 parts contains |V (G)| − 1 vertices, and the last part contains |V (G)| − α (G) − 1 vertices. We then color the edges of F by red and blue such that R = χ (H) − 2 K|V (G)|−1 ∪ K|V (G)|−α (G)−1 + tK1 ,
(
)
(
)
and all the other edges blue. There are two types of components of R: one type is K|V (G)|−1 that contains no G as G is connected, and the other type is K|V (G)|−α (G)−1 + tK1 . The last component of R contains no G as the independence number of any subgraph induced by |V (G)| vertices is greater than α (G). On the other hand, B is a (χ (H) − 1)-partite graph with B = K|V (G)|−1,|V (G)|−1,...,|V (G)|−1,|V (G)|−α (H)+t −1 , then the chromatic number of B is χ (H) − 1. Suppose B contains an H, we have χ (H) ≤ χ (B) = χ (H) − 1, a contradiction. Thus B contains no H, which implies the desired upper bound. If G is H-good, we have that r = (χ (H) − 1)(|V (G)| − 1) + s(H). Then RK (G, H) ≤ α (G) + r − (χ (H) − 1)(|V (G)| − 1) − 1 = α (G) + s(H) − 1. □ We will see that the bound (3) is sharp for many cases when G is H-good. If G is not H-good, the bound (3) does not hold. For example, if one of m and n is odd and n ≥ m ≥ 2, it is shown that R(Sm , Sn ) = m + n in [4,6] and RK (Sm , Sn ) = m + n − 1 in [16]. Thus RK (Sm , Sn ) > α (Sm ) + s(Sn ) − 1 = m. This example tells us that the bound (2) is also sharp for cases that G is not H-good. Lemma 2 ([5]). For any integers m ≥ 2 and n ≥ 1, R(K1,n , Km ) = n(m − 1) + 1. Proof of Theorem 1. The upper bound follows from Lemma 1. For the lower bound, consider the graph G = G1 + G2 with G1 = Kn(m−2)+1 and G2 = nK1 . We will proceed by induction on m. Case 1. m = 2. Then for any n ≥ 2, G = K1,n → (K1,n , K2 ). Case 2. m ≥ 3. Assume the assertion holds for m − 1. For any 2-coloring of the edges of graph G, suppose that G contains no blue Km , and we shall show that G must contain a red K1,n . By induction, G contains a blue Km−1 . If there is a blue Km−1 in G1 , since G contains no blue Km , any vertex in V (G) \ V (Km−1 ) is adjacent to at least one vertex in V (Km−1 ) by a red edge. So |V (G) \ V (Km−1 )| = (n − 1)(m − 1) + 1, then there is a vertex in V (Km−1 ) adjacent to at least n vertices in V (G) \ V (Km−1 ) by red edges, producing a red K1,n . If there is no blue Km−1 in G1 , then the blue Km−1 in G has m − 2 vertices in V (G1 ), and one vertex in V (G2 ), respectively. As there is no blue Km−1 in G1 , by the similar argument, there must be a vertex in Km−2 adjacent to at least n vertices in V (G1 ) \ V (Km−2 ) by red edges, producing a red K1,n . □
4
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
Fig. 1. u1 ∈ V (G1 ), v1 , v2 , u2 ∈ V (G2 ).
3. Paths versus C4 For the figures in this section, red edges are solid and blue edges are dashed. The path Pn and the cycle Cn will be denoted as p1 p2 . . . pn and c1 c2 . . . cn with pi ∈ V (Pn ) and ci ∈ V (Cn ) for 1 ≤ i ≤ n, respectively. Lemma 3 ([3]). For all n ≥ 3, R(Pn , C4 ) = n + 1. Lemma 4 ([12]). For all n ≥ 3, RS (Pn , C4 ) = n − 3. Proof of Theorem 2. For the upper bound, color the edges of K⌊n/2⌋ + (⌈n/2⌉ + 1)K1 red and blue such that the red graph is isomorphic to K⌊n/2⌋−1 + (⌈n/2⌉ + 1)K1 and the blue graph is isomorphic to K1,n . So it contains neither a red path Pn nor a blue cycle C4 , which yields RK (Pn , C4 ) ≤ ⌈n/2⌉. For the lower bound, we will see K⌊n/2⌋+1 + ⌈n/2⌉K1 → (Pn , C4 ) by induction on n. For n = 4, Lemmas 3 and 4 imply K3 + 2K1 → (P4 , C4 ). For n ≥ 5, we consider the parity of n. Case 1. For n even, consider the graph G = G1 + G2 with G1 = Kn/2+1 and G2 = (n/2)K1 . For any vertex u1 ∈ V (G1 ), by induction, we obtain that G \ {u1 } contains a red path Pn−1 , denoted by P with endpoints v1 and v2 . Let u2 be the vertex of V (G) \ (V (P) ∪ {u1 }). Suppose that G contains no red Pn . Subcase 1.1. Let v1 , v2 ∈ V (G1 ). Since G contains no red Pn , the edges u1 v1 , u1 v2 , u2 v1 and u2 v2 are all blue, producing a blue C4 . Subcase 1.2. Let v1 ∈ V (G1 ), v2 ∈ V (G2 ). Then u2 ∈ V (G2 ). Since G contains no red Pn , the edges u1 v1 , u1 v2 and u2 v1 are all blue. Then v1 v2 must be blue, otherwise u1 and u2 would be adjacent to all the vertices in V (P) ∩ V (G1 ) by blue edges, which forms a blue C4 . If u1 u2 is blue, there is a blue C4 : u1 u2 v1 v2 . Assume u1 u2 is red. Let u3 be the vertex adjacent to v2 in P. Since u3 ∈ V (G1 ) and u1 u2 is red, we obtain u1 u3 and u2 u3 are both blue, producing a blue C4 : u1 u3 u2 v1 . Subcase 1.3. Let v1 , v2 ∈ V (G2 ). If u2 ∈ V (G1 ), G contains either a red Pn or a blue C4 , similar to Subcase 1.1. Thus we suppose that u2 ∈ V (G2 ). Since there is at least one vertex of V (G1 ) between any two vertices of V (G2 ) in the path P, by the Pigeonhole Principle, there are three vertices of V (G1 ), denoted by p1 , p2 , p3 , such that p1 p2 p3 ⊆ P, or there are four vertices of V (G1 ), denoted by p′1 , p′2 , p′3 , p′4 , such that p′1 p′2 , p′3 p′4 ⊆ P. If p1 p2 p3 ⊆ P, then v1 p2 and v2 p2 are both blue, otherwise we obtain a new red path on n − 1 vertices with one endpoint in V (G1 ) and the other endpoint in V (G2 ), and it becomes Subcase 1.2. Thus, u1 v1 p2 v2 forms a blue cycle C4 . If p′1 p′2 , p′3 p′4 ⊆ P, v2 p′3 must be blue (see Fig. 1), otherwise we obtain a new red path on n − 1 vertices with endpoints p′4 and v1 . Then v1 p′3 is red, or we can get a blue cycle u1 v1 p′3 v2 . Similarly, v2 p′2 is also red, and thus we get a new red path on n − 1 vertices with endpoints p′1 and p′4 , which becomes Subcase 1.1. ( ) Case 2. For n odd, consider the graph G = G1 + G2 with G1 = K(n+1)/2 and G2 = (n + 1)/2 K1 . For any vertex u1 ∈ V (G2 ), by induction, we obtain that G \ {u1 } contains a red path Pn−1 , denoted by P with endpoints v1 and v2 . Let u2 be the vertex of V (G) \ (V (P) ∪ {u1 }). Suppose that G contains no red Pn . Subcase 2.1. Let v1 , v2 ∈ V (G1 ). Since G contains no red Pn , the edges u1 v1 , u1 v2 , u2 v1 and u2 v2 are all blue, producing a blue C4 . Subcase 2.2. Let v1 ∈ V (G1 ), v2 ∈ V (G2 ). If u2 ∈ V (G1 ), it is similar to Subcase 1.2. Now suppose that u2 ∈ V (G2 ). By the Pigeonhole Principle, there are two vertices of V (G1 ), other than v1 , denoted by p1 , p2 , such that p1 p2 ⊆ P (see Fig. 2). Since G contains no red Pn , the edges u1 v1 , u2 v1 and v1 v2 are blue. And p1 v2 is also blue, otherwise we get a new red path on n − 1 vertices with endpoints p2 and v1 , resulting in Subcase 2.1. If p1 u2 is blue, then we have a blue C4 : u2 v1 v2 p1 . So we suppose that p1 u2 is red, and using the same method, p1 u1 is also red. Then u1 p2 and u2 p2 are both blue, or we can obtain a red path on n vertices by replacing p1 p2 with p1 u1 p2 and p1 u2 p2 , respectively. Thus, we obtain a blue C4 : v1 u1 p2 u2 . Subcase 2.3. Let v1 , v2 ∈ V (G2 ). If u2 ∈ V (G1 ), by the Pigeonhole Principle, there are two vertices of V (G1 ), denoted by p1 , p2 , such that p1 p2 ⊆ P (see Fig. 3). Thus v1 p2 and v2 p1 are blue, or we can obtain a red path on n − 1 vertices with one endpoint in V (G1 ) and the other endpoint in V (G2 ). Since G contains no red Pn , at least one of u1 p1 and u1 p2 is blue, say u1 p2 . Then u1 u2 is red, or we get a blue C4 : v1 u2 u1 p2 . Denote the vertices adjacent to v1 and v2 in path P by p and p′ , and then we get a blue C4 : u1 pu2 p′ .
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
5
Fig. 2. v1 ∈ V (G1 ), v2 , u1 , u2 ∈ V (G2 ).
Fig. 3. u2 ∈ V (G1 ), v1 , v2 , u1 ∈ V (G2 ).
Fig. 4. v1 , v2 , u1 , u2 ∈ V (G2 ).
If u2 ∈ V (G2 ), there are either three vertices of V (G1 ), denoted by p1 , p2 , p3 , such that p1 p2 p3 ⊆ P, or three pairs of vertices of V (G1 ), denoted by p′1 , p′2 , p′3 , p′4 , and p′5 , p′6 , such that p′1 p′2 , p′3 p′4 , p′5 p′6 ⊆ P. If p1 p2 p3 ⊆ P (see Fig. 4), then v1 p2 , v1 p3 , v2 p1 , v2 p2 are all blue. If p2 u1 is blue, then p1 u1 and p3 u1 are both red, or we get a blue C4 . And now we have a new red path on n − 1 vertices by replacing p2 with u1 , which becomes the case where v1 , v2 ∈ V (G2 ) and u2 ∈ V (G1 ). The case will be similar if p2 u2 is blue. Thus, we suppose that p2 u1 and p2 u2 are both red, and then p1 u1 , p3 u1 , p1 u2 , p3 u2 are all blue, producing a blue cycle C4 . If p′1 p′2 , p′3 p′4 , p′5 p′6 ⊆ P (see Fig. 4), v1 p′i and v2 p′j are both blue, for i ∈ {2, 4, 6}, j ∈ {1, 3, 5}, and at least one of p′3 u1 and p′4 u1 is blue. Without loss of generality, suppose that p′3 u1 is blue. Since there is no blue C4 , p′1 u1 and p′5 u1 are red, and p′2 u1 and p′6 u1 are blue, along with p′2 v1 , p′6 v1 , producing a blue C4 . □ 4. Fans versus K3 Before proceeding to the proof, we need some notation and lemmas. If there is a red–blue edge coloring of F such that there is neither a red G nor a blue H, we call such a coloring to be a (G, H)-free coloring and the graph F is called a (G, H)-free graph. In a red–blue edge coloring of F , the subgraph induced by the red edges is denoted by F R and the subgraph induced by the blue edges is denoted by F B . And for a vertex x of F , we denote the numbers of the red and blue neighbors of x in F by degFR (x) and degFB (x), respectively. Lemma 5 ([14]). For all n ≥ 2, R(Fn , K3 ) = 4n + 1. Lemma 6 ([13]). For all n ≥ 2, RS (Fn , K3 ) = 2n − 2. Lemma 7 ([13]). Let n ≥ 2 be an integer and r = R(Fn , K3 ) = 4n + 1. Define a class of red–blue edge colored K4n as H = {H1 , H2 , . . . , H2n }, such that Hi : HiB = K2n,2n \ iK2 HiR = K4n \ HiB , where iK2 is i independent edges of K2n,2n . For any (Fn , K3 )-free coloring of Kr −1 , the resulting graph must belong to the class H.
6
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
Lemma 8 ([15]). For all n ≥ 2, R(nK2 , K3 ) = 2n + 1. Lemma 9 ([13]). Let n ≥ 2 be an integer and r = R(nK2 , K3 ) = 2n + 1. Define a class of red–blue edge colored K2n as H = {H1 , H2 , . . . , H⌈n/2⌉−1 }, such that Hi : HiR = K2i+1 ∪ K2n−2i−1 HiB = K2i+1,2n−2i−1 , where 1 ≤ i ≤ ⌈n/2⌉ − 1. For any (nK2 , K3 )-free coloring of Kr −1 , the resulting graph must belong to the class H. Proof of Theorem 3. The upper bound follows from Lemma 1. For the lower bound, consider the graph G = G1 + G2 with G1 = K3n+1 and G2 = nK1 . We will show that G contains either a red Fn with the central vertex in V (G1 ) or a blue K3 by induction on n. Case 1. Let n = 2. By Lemma 6, we obtain K9 \ K2 → (F2 , K3 ). If G contains no blue K3 , then G must contain a red F2 . Assume that there is no red F2 with the central vertex in V (G1 ), and denote the set of the central vertices of all red F2 in G by V with V ⊆ V (G2 ). Let V (G2 ) = {v1 , v1′ } with v1 ̸ = v1′ . Subcase 1.1. Let V = {v1 } (or V = {v1′ }). Since G \ {v1 } ̸ → (F2 , K3 ), by Lemma 7, G \ {v1 } must belong to the defined class H for n = 2. Since v1 is adjacent to all the vertices in V (G1 ), v1 has at most one red neighbor in each red clique, otherwise we obtain a red F2 with central vertex in V (G1 ). Then v1 has at least two blue neighbors in each red clique, producing a blue K3 . Subcase 1.2. Let V = {v1 , v1′ }. Let F and F ′ be two red F2 in G with V (F ) = {v1 , u1 , u2 , u3 , u4 } and V (F ′ ) = {v1′ , u′1 , u′2 , u′3 , u′4 }, respectively, and ui , u′i ∈ V (G1 ), for 1 ≤ i ≤ 4. Then |V (F ) ∩ V (F ′ )| ≥ 1. If |V (F ) ∩ V (F ′ )| = 1 or |V (F ) ∩ V (F ′ )| = 3, there is a red F2 with central vertex in V (G1 ) trivially. If |V (F ) ∩ V (F ′ )| = 2 and |E(F ) ∩ E(F ′ )| = 0, there is also a red F2 with central vertex in V (G1 ) trivially, and then we assume that |V (F ) ∩ V (F ′ )| = 2 and |E(F ) ∩ E(F ′ )| = 1. Without loss of generality, let u1 = u′1 , u2 = u′2 , and u1 u2 , u3 u4 , u′3 u′4 are red edges in G. Since G contains no red F2 with central vertex in V (G1 ), u2 is adjacent to u3 , u′3 , u4 , u′4 by blue edges in G. And then {u3 , u′3 , u4 , u′4 } induces a red clique in G. Along with the red edges v1 u3 and v1 u4 , we obtain a red F2 with central vertex u3 . If |V (F ) ∩ V (F ′ )| = 4 and |E(F ) ∩ E(F ′ )| = 0, there is also a red F2 with central vertex in V (G1 ) trivially, and then we assume that |V (F ) ∩ V (F ′ )| = 4 and |E(F ) ∩ E(F ′ )| = 2. Without loss of generality, let u1 = u′1 , u2 = u′2 , u3 = u′3 , u4 = u′4 , and u1 u2 , u3 u4 are red edges in G. For any vertex u ∈ V (G1 ) \ {u1 , u2 , u3 , u4 }, there is at least one blue edge in {uu1 , uu2 , uu3 , uu4 }. If uu1 or uu2 is blue, since u1 u3 , u1 u4 , u2 u3 , u2 u4 are all blue, then uu3 and uu4 are both red. Similarly, if uu3 or uu4 is blue, uu1 and uu2 are both red. Since there are three vertices in V (G1 ) \{u1 , u2 , u3 , u4 }, there are at least two vertices, denoted by u and u′ , such that uu1 , uu2 , u′ u1 , u′ u2 or uu3 , uu4 , u′ u3 , u′ u4 are all red. Without loss of generality, we assume that uu1 , uu2 , u′ u1 , u′ u2 are all red. Since G contains no red F2 with central vertex in V (G1 ), uv1 and u′ v1 are both blue. Then uu′ must be red, and {u, u′ , u1 , u2 } induces a red clique in G, along with v1 u1 and v1 u2 , forming a red F2 with central vertex u1 . Thus all the cases produce a red F2 with central vertex in V (G1 ), which contradicts our assumption. Case 2. Let n ≥ 3. Assume the assertion holds for n − 1. If the edges between V (G1 ) and V (G2 ) in G are completely red, then for any vertex x ∈ V (G1 ), degGR (x) ≤ n − 1, 1 otherwise we obtain a red Fn with central vertex x. Then degGB (x) ≥ 2n + 1. Suppose that G contains no blue K3 , and then 1 the blue neighbors of x induce a red clique containing Fn in G1 , a contradiction. Thus there exists at least one blue edge between V (G1 ) and V (G2 ), denoted by the edge u1 v with u1 ∈ V (G1 ) and v ∈ V (G2 ). If degGB (u1 ) = 0, by Lemma 8, G1 contains a red Fn with central vertex u1 or a blue K3 . Then degGB (u1 ) ≥ 1, and we 1 1 denote u2 the vertex adjacent to u1 by a blue edge in G1 . Similarly, we can also find vertex u3 ∈ V (G1 ) \ {u1 , u2 } such that u2 u3 is blue. Since G contains no blue K3 , v u2 and u1 u3 are red. By induction, G \ {v, u1 , u2 , u3 } must contain a red Fn−1 with central vertex u ∈ V (G1 ). Subcase 2.1. Assume uv is red. Since v u2 is red, uu2 must be blue. As u2 u1 and u2 u3 are blue, we have uu1 and uu3 are both red. Thus, V (Fn−1 ) ∪ {u1 , u3 } produces a red Fn in G, a contradiction. Subcase 2.2. Assume uv is blue. Then uu1 is red. Since u1 u3 is red, uu3 is blue. Then v u3 and uu2 are red. Note that u is adjacent to any vertex in V (G) \ (V (Fn−1 ) ∪ {u1 , u2 }) by a blue edge. Otherwise, we obtain either a red Fn with central vertex u or a blue K3 in G. Thus, vertex u has 2n blue neighbors and also has 2n red neighbors. And these blue neighbors induce a completely red graph in G, denoted by Q . If |V (Q ) ∩ V (G1 )| > n, for vertex u′ ∈ V (Q ) ∩ V (G1 ), at least one of u1 u′ and u2 u′ is red. Since |V (Q ) ∩ V (G1 )| ≥ n ≥ 3, one of the vertices u1 and u2 , say u1 , has at least two red neighbors in V (Q ) ∩ V (G1 ). Then we obtain a red Fn with V (Fn ) = {u1 } ∪ V (Q ) and central vertex in V (G1 ). If V (Q ) ∩ V (G1 ) = n, then the blue neighbors of vertex u induce graph Q = Kn + nK1 = Q1 + Q2 in G, and the red neighbors of vertex u induce graph Q ′ = K2n in G. Since Q is completely red, and u3 ∈ Q1 , v ∈ Q2 , it is easily shown that u1 is adjacent to every vertex in V (Q1 ) by a red edge, and u2 is adjacent to every vertex in V (Q2 ) by a red edge, otherwise we can get either a red Fn with central vertex in V (Q1 ) or a blue K3 . If degQB ′ (u2 ) ≥ 2, let u1 and u′ be the blue neighbors of u2 in Q ′ . Similar to u1 , u′ must be adjacent to every vertex in V (Q1 ) by a red edge. Since u1 u′ is red, we obtain a red Fn with
Y. Wang and Y. Li / Discrete Mathematics 343 (2020) 111743
7
central vertex in V (Q1 ) using vertices V (Q ) ∪ {u1 , u′ } \ {v}. Then we assume that degQB ′ (u2 ) = 1. Since degQR ′ (u2 ) = 2n − 2 and Q ′ is (nK2 , K3 )-free, by Lemma 9, Q ′ is the graph H0 in H with degQB ′ (u1 ) = 2n − 1. And by the similar argument above, for any vertex in V (Q ′ ) \ {u1 }, this vertex is adjacent to every vertex in V (Q2 ) by a red edge. As V (Q ′ ) \ {u1 } induces a red K2n−1 in G, along with the red edges between V (Q ′ ) \ {u1 } and V (Q2 ), we obtain a red Fn with central vertex in V (Q ′ ). □ Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors are grateful to the referees for their invaluable comments. Particularly, following their comments for the proof of Lemma 1 (original Theorem 1), we obtain a new bound (2) such that the original bound (3) is a special case of (2). Also, the paragraph after Definition 3 comes from one of reports. References [1] S.A. Burr, A survey of noncomplete Ramsey theory for graphs, Ann. New York Acad. Sci. 328 (1979) 58–75. [2] S.A. Burr, Ramsey numbers involving graphs with long suspended paths, J. Lond. Math. Soc. 24 (1981) 405–413. [3] S.A. Burr, P. Erdős, R.J. Faudree, C.C. Rousseau, R.H. Schelp, Some complete bipartite graph-tree Ramsey numbers, Ann. Discrete Math. 41 (1989) 79–89. [4] S.A. Burr, J. Roberts, On Ramsey numbers for stars, Util. Math. 4 (1973) 217–220. [5] V. Chvátal, Tree-complete graph Ramsey numbers, J. Graph Theory 1 (1977) 93. [6] V. Chvátal, F. Harary, Generalized Ramsey theory for graphs II, small diagonal numbers, Proc. Amer. Math. Soc. 32 (1972) 389–394. [7] P. Erdős, R.J. Faudree, Size Ramsey numbers involving matchings, Finite Infin. Sets 37 (1984) 247–264. [8] P. Erdős, R.J. Faudree, C. Rousseau, R. Schelp, The size Ramsey numbers, Period. Math. Hungar. 9 (1978) 145–161. [9] R.J. Faudree, R.H. Schelp, A survey of results on the size Ramsey number, in: Paul Erdös and His Mathematics II(Budapest, 1999), in: Bolyai Soc. Math. Stud., vol. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 291–309. [10] S. Haghi, H. Maimani, A. Seify, Star-critical Ramsey numbers of Fn versus K4 , Discrete Appl. Math. 217 (2017) 203–209. [11] Y. Hao, Q. Lin, Star-critical Ramsey numbers for large generalized fans and books, Discrete Math. 341 (2018) 3385–3393. [12] J. Hook, G. Isaak, Star-critical Ramsey numbers, Discrete Appl. Math. 159 (2011) 328–334. [13] Z. Li, Y. Li, Some star-critical Ramsey numbers, Discrete Appl. Math. 181 (2015) 301–305. [14] Y. Li, C. Rousseau, Fan-complete graph Ramsey numbers, J. Graph Theory 23 (1996) 413–420. [15] P. Lorimer, The Ramsey numbers for stripes and one complete graph, J. Graph Theory 8 (1984) 177–184. [16] Y. Wang, Y. Li, Redundant edges in Ramsey graphs, Discrete Appl. Math., revised. [17] D.B. West, Introduction to Graph Theory, second ed., Prentice Hall, New Jersey, 2000. [18] Y. Wu, Y. Sun, S. Radziszowski, Wheel and star-critical Ramsey numbers for quadrilateral, Discrete Appl. Math. 186 (2015) 260–271. [19] Y. Zhang, H. Broersma, Y. Chen, On star-critical and upper size Ramsey numbers, Discrete Appl. Math. 202 (2016) 174–180.