Ramsey-goodness -- and otherwise

TL;DR

This paper investigates the Ramsey numbers of graphs with bounded degree and bandwidth, showing that bandwidth restrictions can significantly improve bounds and that Burr's conjecture does not hold universally.

Contribution

It demonstrates that bandwidth constraints can reduce Ramsey number bounds and provides counterexamples to Burr's conjecture for certain graph classes.

Findings

Bandwidth restrictions lead to linear bounds on Ramsey numbers.

Counterexamples show Burr's conjecture fails for some graphs.

Ramsey numbers for graphs with bounded bandwidth are explicitly characterized.

Abstract

A celebrated result of Chv\'atal, R\"odl, Szemer\'edi and Trotter states (in slightly weakened form) that, for every natural number $\Delta$, there is a constant $r_\Delta$ such that, for any connected $n$-vertex graph $G$ with maximum degree $\Delta$, the Ramsey number $R(G,G)$ is at most $r_\Delta n$, provided $n$ is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take $r_\Delta = \Delta$. However, Graham, R\"odl and Ruci\'nski showed, by taking $G$ to be a suitable expander graph, that necessarily $r_\Delta > 2^{c\Delta}$ for some constant $c>0$. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of $G$ be at most some function $\beta (n) = o(n)$, then $R(G,G) \le (2\chi(G)+4)n\leq (2\Delta+6)n$, i.e., $r_\Delta = 2\Delta +6$ suffices. On the other hand, we show that Burr's conjecture itself fails even for $P_n^k$, the $k$th power of a path $P_n$. Brandt showed that for any $c$, if $\Delta$ is sufficiently large, there are connected $n$-vertex graphs $G$ with $\Delta(G)\leq\Delta$ but $R(G,K_3)>cn$. We show that, given $\Delta$ and $H$, there are $\beta>0$ and $n_0$ such that, if $G$ is a connected graph on $n\ge n_0$ vertices with maximum degree at most $\Delta$ and bandwidth at most $\beta n$, then we have $R(G,H)=(\chi(H)-1)(n-1)+\sigma(H)$, where $\sigma(H)$ is the smallest size of any part in any $\chi(H)$-partition of $H$. We also show that the same conclusion holds without any restriction on the maximum degree of $G$ if the bandwidth of $G$ is at most $\epsilon(H) \log n/\log\log n$.