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Title: Polarity graphs and ramsey numbers for C4 versus stars
Authors: Zhang, X
Chen, Y
Cheng, TCE 
Keywords: Finite field
Polarity graph
Ramsey number
Issue Date: 2017
Publisher: North-Holland
Source: Discrete mathematics, 2017, v. 340, no. 4, p. 655-660 How to cite?
Journal: Discrete mathematics 
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) shows that R(C4,K1,n)≤n+⌊n−1⌋+2 and if n is the square of a prime power, then the equality holds. In this paper, by discussing the properties of polarity graphs whose vertices are points in the projective planes over Galois fields, we prove that R(C4,K1,q2−t)=q2+q−(t−1) if q is an odd prime power, 1≤t≤2⌈q4⌉ and t≠2⌈q4⌉−1, which extends a result on R(C4,K1,q2−t) obtained by Parsons (1976).
ISSN: 0012-365X
EISSN: 1872-681X
DOI: 10.1016/j.disc.2016.12.005
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