Three-color Ramsey number of an odd cycle versus bipartite graphs with small bandwidth

TL;DR

This paper establishes asymptotically tight bounds on the three-color Ramsey numbers involving odd cycles and bipartite graphs with small bandwidth, extending known results and introducing new classes of graphs.

Contribution

It proves that for balanced bipartite graphs with small bandwidth and bounded degree, the three-color Ramsey number with two copies of the graph and an odd cycle is at most roughly three times the cycle length, generalizing previous results.

Findings

Upper bound of (3+γ)n for large odd n

Exact asymptotics for Ramsey numbers of certain bipartite graphs

Extension of known results to new graph classes

Abstract

A graph $\mathcal{H}=(W,E_\mathcal{H})$ is said to have {\em bandwidth} at most $b$ if there exists a labeling of $W$ as $w_1,w_2,\dots,w_n$ such that $|i-j|\leq b$ for every edge $w_iw_j\in E_\mathcal{H}$. We say that $\mathcal{H}$ is a {\em balanced $(\beta,\Delta)$-graph} if it is a bipartite graph with bandwidth at most $\beta |W|$ and maximum degree at most $\Delta$, and it also has a proper 2-coloring $\chi :W\rightarrow[2]$ such that $||\chi^{-1}(1)|-|\chi^{-1}(2)||\leq\beta|\chi^{-1}(2)|$. In this paper, we prove that for every $\gamma>0$ and every natural number $\Delta$, there exists a constant $\beta>0$ such that for every balanced $(\beta,\Delta)$-graph $\mathcal{H}$ on $n$ vertices we have $$R(\mathcal{H}, \mathcal{H}, C_n) \leq (3+\gamma)n$$ for all sufficiently large odd $n$. The upper bound is sharp for several classes of graphs. Let $\theta_{n,t}$ be the graph consisting of $t$ internally disjoint paths of length $n$ all sharing the same endpoints. As a corollary, for each fixed $t\geq 1$, $R(\theta_{n, t},\theta_{n, t}, C_{nt+\lambda})=(3t+o(1))n,$ where $\lambda=0$ if $nt$ is odd and $\lambda=1$ if $nt$ is even. In particular, we have $R(C_{2n},C_{2n}, C_{2n+1})=(6+o(1))n$, which is a special case of a result of Figaj and {\L}uczak (2018).