An on-line competitive algorithm for coloring bipartite graphs without long induced paths

TL;DR

This paper introduces a new online competitive coloring algorithm specifically designed for bipartite graphs that do not contain long induced paths, advancing the understanding of online coloring in restricted graph classes.

Contribution

The paper presents the first online competitive coloring algorithm for P9-free bipartite graphs, addressing a longstanding open problem in graph coloring.

Findings

Successfully developed an online coloring algorithm for P9-free bipartite graphs

Achieved competitive ratio improvements over previous partial results

Extended the class of bipartite graphs with known online coloring solutions

Abstract

The existence of an on-line competitive algorithm for coloring bipartite graphs remains a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose a new on-line competitive coloring algorithm for $P_9$-free bipartite graphs.