Odd Induced Subgraphs in Graphs with Treewidth at Most Two

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 (Comb Probab Comput 1:335–349, 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 (Discrete Math 275–279, 1995) proved $c=\frac{2}{3}$ for trees, Berman et al. (Aust J Comb 81–85, 1997) showed that $c=\frac{2}{5}$ 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.