Odd induced subgraphs in graphs with treewidth at most two

TL;DR

This paper confirms a conjecture about the existence of large induced subgraphs with all odd degrees in graphs with treewidth at most two, establishing the optimal constant for this class.

Contribution

It proves the conjecture for graphs with treewidth at most two and determines the exact optimal constant c=2/5 for this family.

Findings

Confirmed the conjecture for graphs with treewidth at most two.

Established the optimal constant c=2/5 for this class.

Bound is proven to be best possible.

Abstract

A long-standing conjecture asserts that there exists a constant $c>0$ such that every graph of order $n$ without isolated vertices contains an induced subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that every graph $G$ has an induced subgraph of order at least $|V(G)|/(2\chi(G))$ with all degrees odd, where $\chi(G)$ is the chromatic number of $G$, this implies the conjecture for graphs with { bounded} chromatic number. But the factor $1/(2\chi(G))$ seems to be not best possible, for example, Radcliffe and Scott (1995) proved $c=\frac 23$ for trees, Berman, Wang and Wargo (1997) showed that $c=\frac 25$ for graphs with maximum degree $3$, so it is interesting to determine the exact value of $c$ for special family of graphs. In this paper, we further confirm the conjecture for graphs with treewidth at most 2 with $c=\frac{2}{5}$, and the bound is best possible.