List $3$-coloring $C_4$-free graphs of diameter-$2$ in polynomial-time

Yukihiro Murakami

List $3$-coloring $C_4$-free graphs of diameter-$2$ in polynomial-time

Combinatorics 2026-06-29 v1 Discrete Mathematics

Authors: Yukihiro Murakami

Abstract

We show that list $3$ -coloring a~ $C_4$ -free graph of diameter- $2$ can be done in polynomial-time. Our algorithm is based on a structural characterization showing that many such graphs are not~ $3$ -colorable. In particular, we show that~ $C_4$ -free graphs of diameter- $2$ without universal vertices, where the maximum degree is at least~ $17$ , are not~ $3$ -colorable.

Keywords

list coloring C4-free graphs diameter-2 polynomial-time algorithm structural characterization

Cite

@article{arxiv.2606.30282,
  title  = {List $3$-coloring $C_4$-free graphs of diameter-$2$ in polynomial-time},
  author = {Yukihiro Murakami},
  journal= {arXiv preprint arXiv:2606.30282},
  year   = {2026}
}

Comments

15 pages, 3 figures