Maximum odd induced subgraph of a graph concerning its chromatic number

TL;DR

This paper investigates the maximum size of odd induced subgraphs in relation to a graph's chromatic number, disproving some existing conjectures and confirming others for specific graph classes.

Contribution

It disproves Scott's conjecture for bipartite graphs, confirms it for line graphs, and disapproves a conjecture by Berman, Wang, and Wargo.

Findings

Scott's conjecture is false for bipartite graphs.

Scott's conjecture holds for all line graphs.

A conjecture by Berman, Wang, and Wargo is disproved.

Abstract

Let $f_{o}(G)$ be the maximum order of an odd induced subgraph of $G$. In 1992, Scott proposed a conjecture that $f_{o}(G)\geq \frac {n} {2\chi(G)}$ for a graph $G$ of order $n$ without isolated vertices, where $\chi(G)$ is the chromatic number of $G$. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang and Wargo in 1997, which states that $f_{o}(G)\geq 2\lfloor\frac {n} {4}\rfloor$ for a connected graph $G$ of order $n$. Scott's conjecture is open for a graph with chromatic number at least 3.