Year: 2026
Type: preprint
Abstract: Let \(B_n\) denote the book graph consisting of \(n\) triangles sharing a common edge. Rousseau and Sheehan (1978) proved, using Paley graphs, that \(R(B_n,B_n)=4n+2\) whenever \(4n+1\) is a prime power. Since then, only a few additional exact values of \(R(B_n,B_n)\) have been obtained. In this paper, we obtain \(R(B_n,B_n)=4n+1\) for infinitely many \(n\) by constructing graphs from pseudo-cyclic strongly regular graphs (PC-graphs). To the best of our knowledge, this is the first new infinite family of exact diagonal book Ramsey numbers since 1978. We also prove that \(R(B_{n-2},B_n)\le 4n-3\) for every \(n\ge 3\) with \(n\ne 6\), removing the original condition \(n\equiv 2\pmod 3\) imposed by Rousseau and Sheehan. Moreover, equality \(R(B_{n-2},B_n)=4n-3\) holds if there exists a symmetric Hadamard matrix of order \(2n-2\) with all diagonal entries equal to \(1\). In particular, this equality holds for every \(n=2^{2\ell-1}+1\) with \(\ell\ge 1\).
Source: arXiv (Cornell University)
Authors: Dai, Lulu, Lin, Qizhong
Institution: Fuzhou University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7163887635