On a problem of Sivaraman and a problem of Gyárfás

Abstract

The \textit{girth} of a graph $G$, denoted $\mathrm{g}(G)$, is the length of a shortest cycle in $G$. If $G$ contains no cycle, we define $\mathrm{g}(G)=\infty$. Sivaraman (2020) asked for the optimal $χ$-bounding function for the class of graphs whose complements have girth at least $6$. Let \(F(s) = \max\{χ(G): ω(G)\le s,\ \mathrm{g}(\overline{G})\ge 6\}\). We prove that there exists a constant \(c>0\) such that \[ c\left(\frac{s}{\log s}\right)^{4/3} \le F(s) \le (1+o(1))\frac{s^{3/2}}{\log s}. \] For small values, we establish the exact results \[ F(1)=1,\; F(2)=2,\; F(3)=4,\; F(4)=5,\; F(5)=6,\; F(6)=8, \] and each bound is sharp. A graph $G$ is \emph{almost perfect} if every induced subgraph $H$ of $G$ satisfies \(α(H)ω(H)+1\ge |V(H)|\). Gyárfás (2023) asked whether almost perfect graphs are $χ$-bounded by the function $g(x)=x+1$. We answer this question in the negative by showing that there is no constant $c$ such that every almost perfect graph $G$ satisfies $χ(G)\le ω(G)+c$.