Coloring $(P_6,C_4)$-free graphs with $Δ- 1$ colors

Uttam K. Gupta; Dinabandhu Pradhan; Rashmi Rekha Swain

Coloring $(P_6,C_4)$-free graphs with $Δ- 1$ colors

Combinatorics 2026-07-14 v1 Discrete Mathematics

Authors: Uttam K. Gupta , Dinabandhu Pradhan , Rashmi Rekha Swain

Abstract

For a graph $G$ , let $\Delta(G)$ , $\omega(G)$ , and $\chi(G)$ denote the maximum degree, clique number, and chromatic number of $G$ , respectively. Let $P_n$ and $C_n$ denote the chordless path and chordless cycle on $n$ vertices, respectively. In this paper, we prove that every $(P_6,C_4)$ -free graph $G$ with $\Delta(G)\ge 9$ and $\omega(G)<\Delta(G)$ is $(\Delta(G)-1)$ -colorable.

Keywords

graph coloring chromatic number maximum degree clique number forbidden induced subgraphs P6-free graphs C4-free graphs

Cite

@article{arxiv.2607.12367,
  title  = {Coloring $(P_6,C_4)$-free graphs with $Δ- 1$ colors},
  author = {Uttam K. Gupta and Dinabandhu Pradhan and Rashmi Rekha Swain},
  journal= {arXiv preprint arXiv:2607.12367},
  year   = {2026}
}