Year: 2026
Type: preprint
Abstract: By the Strong Perfect Graph Theorem, every graph whose chromatic number exceeds its clique number contains an odd hole or an odd antihole. We investigate the structure of odd antiholes in critical graphs with fixed maximum degree. We prove that if $G$ is a $k$-critical graph with maximum degree $Δ$ then every odd antihole in $G$ has length at most $2(Δ-k)+7$, and conjecture that, in the odd-hole-free setting, equality forces the graph itself to be the corresponding odd antihole. We verify this extremal characterization when $Δ\geq 2k-5$, extending recent results for $Δ=k$, and construct $k$-critical odd-hole-free graphs showing that the next possible odd antihole length $2(Δ-k)+5$ below the extremal bound can occur. Our proofs use a list-coloring theorem for graphs on $2\ell-1$ vertices whose complements are hypomatchable. This motivates the conjecture that every odd antihole $\overline{C_{2\ell-1}}$ is strongly $\ell$-chromatic-choosable; we show that this conjecture would imply the full extremal characterization. We also obtain degree and attachment restrictions for odd antiholes in critical graphs.
Source: arXiv (Cornell University)
Authors: Fiujlaali, Alireza, McDonald, Jessica
Institutions: Georgia State University, Auburn University
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7221823885