Claw-free perfect graphs are chromatic-choosable
Abstract
We prove that every claw-free perfect graph G satisfies ch(G)=χ(G)=ω(G). More strongly, if S is a clique of simplicial vertices of G and k≥ω(G), every proper precoloring of S extends to a proper coloring of G from arbitrary lists of size k on V(G)∖ S. The proof has three coloring steps: a mixed-list theorem for cobipartite graphs, a precoloring-extension theorem for elementary graphs built on Galvin's kernel method and a homogeneous-pair exchange argument, and a polynomial method (Combinatorial Nullstellensatz) argument for graphs of interval type in which a Vandermonde factor is transferred across each incomplete pair. The three coloring steps are assembled along the structure theorems of Chvátal--Sbihi, Maffray--Reed and Chudnovsky--Seymour, using double clique cutsets for composition. As a consequence, every perfect line graph is chromatic-choosable, recovering the theorem of Peterson and Woodall. Our precoloring extension theorem also resolves a conjecture posed by Chudnovsky and Seymour.
Related
BibTeX
@misc{mishra2026clawfree,
title = {{Claw-free perfect graphs are chromatic-choosable}},
author = {Suchismita Mishra and Narayanan Narayanan and Umesh Shankar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01068},
eprint = {2611.01068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01068},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Claw-free perfect graphs are chromatic-choosable
AU - Suchismita Mishra
AU - Narayanan Narayanan
AU - Umesh Shankar
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01068
UR - https://arxiv.org/abs/2611.01068
AB - We prove that every claw-free perfect graph G satisfies ch(G)=χ(G)=ω(G). More strongly, if S is a clique of simplicial vertices of G and k≥ω(G), every proper precoloring of S extends to a proper coloring of G from arbitrary lists of size k on V(G)∖ S. The proof has three coloring steps: a mixed-list theorem for cobipartite graphs, a precoloring-extension theorem for elementary graphs built on Galvin's kernel method and a homogeneous-pair exchange argument, and a polynomial method (Combinatorial Nullstellensatz) argument for graphs of interval type in which a Vandermonde factor is transferred across each incomplete pair. The three coloring steps are assembled along the structure theorems of Chvátal--Sbihi, Maffray--Reed and Chudnovsky--Seymour, using double clique cutsets for composition. As a consequence, every perfect line graph is chromatic-choosable, recovering the theorem of Peterson and Woodall. Our precoloring extension theorem also resolves a conjecture posed by Chudnovsky and Seymour.
ER -