On Scott's odd induced subgraph conjecture and a related problem

TL;DR

This paper investigates Scott's odd induced subgraph conjecture, confirming it for claw-free graphs, providing counterexamples for certain classes, and analyzing the conjecture's validity for line graphs of regular graphs.

Contribution

It proves Scott's conjecture for claw-free graphs, constructs counterexamples for $K_{1,r}$-free graphs, and examines the conjecture's validity for line graphs of regular graphs.

Findings

Confirmed Scott's conjecture for claw-free graphs without isolated vertices.

Constructed $K_{1,r}$-free graphs where the conjecture fails for $r \,\geq\, 4$.

Showed $C_5$ as the smallest counterexample to a related problem.

Abstract

For a graph $G$, let $f_o(G)$ denote the maximum order of an induced subgraph of $G$ all of whose vertices have odd degree, and let $\chi(G)$ denote the chromatic number of $G$. Scott (CPC, 1992) proved that $f_o(G) \ge |V(G)|/(2\chi(G))$ for every graph without isolated vertices, and conjectured that the factor $2$ can be removed. Wang and Wu (JGT, 2024) showed that this conjecture fails for bipartite graphs, but holds for line graphs. In this article, we confirm Scott's conjecture for claw-free graphs without isolated vertices, thereby strengthening the result of Wang and Wu. We also construct $K_{1,r}$-free graphs of arbitrarily large order to show that the conjecture fails for this broader class, for every integer $r \ge 4$. Wang and Wu also asked whether $f_o(L(G)) \ge n/2$ holds for every connected regular graph $G$ of order $n \ge 3$. We show that $C_5$ is the smallest counterexample to this problem. On the positive side, we prove that if $G$ is a connected $k$-regular $C_5$-free graph on $n$ vertices with $k \ge 2$, then $f_o(L(G)) \ge n/2$.