Odd Induced Subgraphs in Graphs with Treewidth at Most Two

A long-standing conjecture asserts that there exists a constant 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 with all degrees odd, where is the chromatic number of G, this implies the conjecture for graphs with bounded chromatic number. But the factor seems to be not best possible, for example, Radcliffe and Scott (Discrete Math 275-279, 1995) proved for trees, Berman et al. (Aust J Comb 81-85, 1997) showed that 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 , and the bound is best possible.