P_4-Colorings and P_4-Bipartite Graphs

Chinh T Hoàng,Van Bang Le

doi:10.46298/dmtcs.272

Abstract

A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.

Highlights

A graph G is perfect if, for each induced subgraph H of G, the chromatic number of H is equal to the clique number of H
Nowadays this conjecture is known as the Strong Perfect Graph Conjecture (SPGC) and is still open
Given a graph G and a positive integer k, a map π : V (G) → {1, . . . , k} is a perfect k-coloring of G if the subgraphs induced by each color class π−1(i) is perfect

Summary

Introduction

A graph G is perfect if, for each induced subgraph H of G, the chromatic number of H is equal to the clique number of H. Our discussion is motivated by the fact that the perfection of a graph depends only on the structure of its induced paths on four vertices (denoted by P4); see [36]. In this sense, graphs with empty P4-structure (P4-free graphs) form a somewhat based graph class in discussing graph’s perfection; they are perfect by a result due to Seinsche [38] (see Jung [31]). We call a perfect k-coloring of a graph P4-free k-coloring if the subgraphs of that graph induced by the color classes are P4-free. Graphs without odd holes and odd antiholes are called Berge graphs

NP-completeness results

Examples of P4-bipartite graphs

Which P4-bipartite graphs are perfect?

P4-bipartite graphs and the SPGC