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Title: Some values of ramsey numbers for C4 versus stars
Authors: Zhang, X
Chen, Y
Cheng, TCE 
Keywords: Finite fields
Ramsey graph
Ramsey number
Issue Date: 2017
Publisher: Academic Press
Source: Finite fields and their applications, 2017, v. 45, p. 73-85 How to cite?
Journal: Finite fields and their applications 
Abstract: For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that for any graph of order N, either G contains a copy of G1 or its complement contains a copy of G2. Let Cm be a cycle of length m and K1,n a star of order n+1. Parsons (1975) [6] shows that R(C4,K1,n)≤n+⌊n−1⌋+2 for all n≥2 and the equality holds if n is the square of a prime power. Let q be a prime power. In this paper, we first construct a graph Γq on q2−1 vertices without C4 by using the Galois field Fq, and then we prove that R(C4,K1,(q−1)2+t)=(q−1)2+q+t for q≥4 is even and t=1,0,−2, and R(C4,K1,q(q−1)−t)=q2−t for q≥5 is odd and t=2,4.,2⌈q4⌉.
ISSN: 1071-5797
EISSN: 1090-2465
DOI: 10.1016/j.ffa.2016.11.012
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