3-Coloring $P_t$-Free Graphs With Only One Prescribed Induced Odd Cycle Length

TL;DR

This paper presents a polynomial-time algorithm for 3-coloring graphs that are free of long paths and have only one specified odd cycle length, specifically for graphs in the class ,7.

Contribution

It introduces a polynomial-time algorithm for 3-coloring ,7 graphs, addressing a specific class of graphs with constrained induced odd cycles.

Findings

Polynomial-time 3-coloring algorithm for ,7 graphs

Characterization of graphs with a single prescribed odd cycle length

Extension of coloring algorithms to restricted graph classes

Abstract

A graph is $P_t$-free if it contains no induced subgraph isomorphic to a $t$-vertex path. A graph is not bipartite if and only if it contains an induced subgraph isomorphic to a $k$-vertex cycle, where $k$ is odd. We focus on the 3-coloring problem for $P_t$-free graphs that have only one prescribed induced odd cycle length. For any integer $t$ and any odd integer $k$, let $\mathcal{G}_{t,k}$ be the class of graphs that are $P_{t}$-free and all their induced odd cycles must be $C_k$. In this paper, we present a polynomial-time algorithm that solves the 3-coloring problem for any graph in $\mathcal{G}_{10,7}$.