Integer round-up property for the chromatic number of some h-perfect graphs

TL;DR

This paper proves that certain classes of graphs have the integer round-up property for their chromatic number, enabling polynomial-time computation of weighted chromatic numbers and contributing to graph coloring conjectures.

Contribution

It establishes the integer round-up property for h-perfect line-graphs and t-perfect claw-free graphs, providing new polynomial-time algorithms for weighted chromatic number calculation.

Findings

Integer round-up property holds for h-perfect line-graphs.

Integer round-up property holds for t-perfect claw-free graphs.

Results support a case of Goldberg and Seymour's edge-coloring conjecture.

Abstract

A graph is h-perfect if its stable set polytope can be completely described by non-negativity, clique and odd-hole constraints. It is t-perfect if it furthermore has no clique of size 4. For every graph $G$ and every $c\in\mathbb{Z}_{+}^{V(G)}$, the weighted chromatic number of $(G,c)$ is the minimum cardinality of a multi-set $\mathcal{F}$ of stable sets of $G$ such that every $v\in V(G)$ belongs to at least $c_v$ members of $\mathcal{F}$. We prove that every h-perfect line-graph and every t-perfect claw-free graph $G$ has the integer round-up property for the chromatic number: for every non-negative integer weight $c$ on the vertices of $G$, the weighted chromatic number of $(G,c)$ can be obtained by rounding up its fractional relaxation. In other words, the stable set polytope of $G$ has the integer decomposition property. Our results imply the existence of a polynomial-time algorithm which computes the weighted chromatic number of t-perfect claw-free graphs and h-perfect line-graphs. Finally, they yield a new case of a conjecture of Goldberg and Seymour on edge-colorings.