Coloring of graphs without long odd holes
Ran Chen, Baogang Xu
TL;DR
The paper studies the colorability of graphs that avoid long odd holes by examining three graph families ${\cal A}_{\ell}$, ${\cal B}_{\ell}$, and ${\cal G}_{\ell}$. It advances two main results: (i) every graph in ${\cal G}_2$ is colorable with at most $1456$ colors and every graph in ${\cal A}_{3}$ is 4-colorable, and (ii) every $7$-hole-free graph in ${\cal B}_{\ell}$ is colorable with at most $12\ell+8$ colors. The proofs deploy advanced structural tools—levellings, stable levellings, lollipops, and pillar-based arguments—to bound local chromatic numbers and propagate these bounds to obtain global color bounds, while carefully excluding long odd holes to reach contradictions when necessary. These results provide explicit colorability bounds in regimes where previous work gave only asymptotic or large constants, refining the understanding of how girth, hole structure, and triangle-freeness influence colorability in sparse graphs.
Abstract
A {\em hole} is an induced cycle of length at least 4, a $k$-hole is a hole of length $k$, and an {\em odd hole} is a hole of odd length. Let $\ell\ge 2$ be an integer. Let ${\cal A}_{\ell}$ be the family of graphs of girth at least $2\ell$ and having no odd holes of length at least $2\ell+3$, let ${\cal B}_{\ell}$ be the triangle-free graphs which have no 5-holes and no odd holes of length at least $2\ell+3$, and let ${\cal G}_{\ell}$ be the family of graphs of girth $2\ell+1$ and have no odd hole of length at least $2\ell+5$. Chudnovsky {\em et al.} \cite{CSS2016} proved that every graph in ${\cal A}_{2}$ is 58000-colorable, and every graph in ${\cal B}_{\ell}$ is $(\ell+1)4^{\ell-1}$-colorable. Lan and liu \cite{LL2023} showed that for $\ell\geq3$, every graph in ${\cal G}_{\ell}$ is 4-colorable. It is not known whether there exists a small constant $c$ such that graphs of ${\cal G}_2$ are $c$-colorable. In this paper, we show that every graph in ${\cal G}_2$ is 1456-colorable, and every graph in ${\cal A}_{3}$ is 4-colorable. We also show that every 7-hole free graph in ${\cal B}_{\ell}$ is $(12\ell+8)$-colorable.