Sufficient conditions for graphs with path-chromatic number 2

Let G be a graph and (T, B) be a tree-decomposition of G, where B = (B t ) t∈V (T ) .The chromatic number of the tree-decomposition (T, B) is the maximum chromatic number among the subgraphs induced by B t .Seymour (2016) defined the tree-chromatic number of a graph G to be the minimum chromatic number of the tree-decompositions of G.In this paper, we focus on sufficient conditions for graphs with tree-chromatic number at most 2.These conditions guarantee the existence of a "tree-like decomposition" into bipartite graphs.We prove that every graph in the following graph classes has tree-chromatic number at most 2; (i) the graphs with maximum degree at most 4 without cycles of length 3 and 5, (ii) the triangle-free graphs with maximum degree at most 3, and (iii) triangle-free planar graphs.More precisely, we only consider path-decompositions of graphs.