On Some Ramsey Numbers for Quadrilaterals Versus Wheels

For given graphs G1 and G2, the Ramsey number R(G1, G2) is the least integer n such that every 2-coloring of the edges of Kn contains a subgraph isomorphic to G1 in the first color or a subgraph isomorphic to G2 in the second color. Surahmat et al. proved that the Ramsey number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R(C_4, W_n) \leq n + \lceil (n-1)/3\rceil}$$\end{document}. By using asymptotic methods one can obtain the following property: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R(C_4, W_n) \leq n + \sqrt{n}+o(1)}$$\end{document}. In this paper we show that in fact \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R(C_4, W_n) \leq n + \sqrt{n-2}+1}$$\end{document} for n ≥ 11. Moreover, by modification of the Erdős-Rényi graph we obtain an exact value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R(C_4, W_{q^2+1}) = q^2 + q + 1}$$\end{document} with q ≥ 4 being a prime power. In addition, we provide exact values for Ramsey numbers R(C4, Wn) for 14 ≤ n ≤ 17.


Introduction
In this paper all graphs considered are undirected, finite and contain neither loops nor multiple edges. Let G be such a graph. The vertex set of G is denoted by V (G), the edge set of G by E(G), and the number of edges in G by e(G). Let d(v) be the degree of vertex v, and let d 1 (v) and d 2 (v) denote the number of the edges incident to v colored with the first and the second color, respectively. By δ i (G) we denote the minimum degree of G in color i. The open neighborhood in color i of vertex v in graph G is N i (v) = {u ∈ V (G)|{u, v} ∈ E(G)and{u, v} is colored with colori}. Define G[S] to be the subgraph of G induced by the set of vertices S ⊂ V (G). Let P n (resp. C n ) be the path (resp. cycle) on n vertices. A wheel W n is a graph on n vertices obtained from a C n−1 by adding one vertex w and making w adjacent to all vertices of the C n−1 .
For given graphs G 1 , G 2 , the Ramsey number R(G 1 , G 2 ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 2 colors, then it always contains a monochromatic copy of G 1 colored with the first color or a monochromatic copy of G 2 colored with the second color. A coloring of the edges of n-vertex complete graph with 2 colors is called a (G 1 , G 2 ; n)-coloring if it does not contain a subgraph isomorphic to G 1 colored with the first color nor a subgraph isomorphic to G 2 colored with the second color.
The Turán number t (n, G) is the maximum number of edges in any n-vertex graph which does not contain a subgraph isomorphic to G. A graph on n vertices is said to be extremal with respect to G if it does not contain a subgraph isomorphic to G and has exactly t (n, G) edges.
Some well known theorems will be used to prove the main result of this paper.

Main Theorem
Proof (Theorem 4) For simplicity of notation, we set k = √ n − 2 . Let us consider a graph G = K n+k+1 and its decomposition consists of all edges of G in ith color. Suppose that for graph G there is a (C 4 , W n ; n + k + 1)-coloring and let us consider such coloring.
First let us assume that there is In this case the minimum possible number of edges in color 1 in G is The last case to consider is δ 1 (G) = k + 1. In this case G 1 has at most t (n + k + 1, Similarly to the previous case let us consider integer p such that n ∈ {( p − 1) 2 + 2, · · · , p 2 + 1}.
We have the following depending on the parity of δ 1 (G) and (n + k + 1).
Proof If δ 1 (G) and |V (G)| = (n + k + 1) are odd, then it is impossible that for all vertices w ∈ V (G) we have d 1 (w) = δ 1 (G). In the worst situation, when all A edges are adjacent to v 1 or v 2 , we have that d 1 We will prove that d 2 (v 1 )+d 2 (v 2 ) ≥ n −1 for all vertices v 1 , v 2 ∈ V (G ) such that {v 1 , v 2 } ∈ E(G 1 ). In this case we obtain a contradiction because by Ore's Theorem subgraph G contains a C n−1 and G contains a W n in the second color.
The remaining part of the proof is divided into three parts.
The exact values of t (n, C 4 ) are known for all n ≤ 21, see [1]. In addition, this paper covers all extremal graphs. Table 1 contains all values needed to prove the inequality One can see that for all 11 ≤ n ≤ 15 the proof is complete. for t (n, C 4 ) are known for all 22 ≤ n ≤ 31, see [11]. Table 2 presents all values needed to finish the checking the inequality d 2 (v 1 ) + d 2 (v 2 ) ≥ n − 1 for 18 ≤ n ≤ 26. We will mark with − the case when A < 0.

n ≥ 27
In this case p ≥ 6. We have that in G d 1 In order to finish the proof we have to show that 2n −  Taking n = q 2 + 1 in Theorem 4, we have Corollary 5 For all integers q, q ≥ 4 R(C 4 , W q 2 +1 ) ≤ q 2 + q + 1.

Erdős-Rényi Graph
Let q be a prime power. The famous Erdős-Rényi graph E R(q), first constructed by Erdős and Rényi in 1962, was studied in detail by Parsons in [4]. We know the following properties of E R(q) : -E R(q) has q 2 + q + 1 vertices, q + 1 vertices with degree q and q 2 vertices with degree q + 1 -E R(q) does not contain a subgraph C 4 -in E R(q) there are no two adjacent vertices of degree q -in E R(q) no vertex of degree q belongs to a subgraph K 3 Let H (q) denote the subgraph of E R(q) obtained by deleting one vertex of degree q. By the third property of E R(q), the subgraph H (q) contains 2q vertices with degree q and q 2 − q vertices with degree q + 1. One can observe that for all vertices w, the degree d(w) in the complement of H (q) is at most q 2 −1. By this fact, the complement of H (q) does not contain a W q 2 +1 , so there exists a (C 4 , W q 2 +1 ; q 2 + q)-coloring. By this fact and by Corollary 5 we have the following Theorem 6 For q ≥ 4 being a prime power R(C 4 , W q 2 +1 ) = q 2 + q + 1.

Exact Values for Small Wheels
Up to date values for R(C 4 , W n ) are known only for n ≤ 13. We determined the next four values as follows: Proof By Theorem 6 we immediately obtain R(C 4 , W 17 ) = 21. In order to determine an upper bound for all remaining cases we use Theorem 4. For a lower bound we present appropriate matrix of critical coloring (see Fig. 1). These matrices were obtained by using simulated annealing to find C 4 -free graphs with a minimum degree 4. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.