Path-Bicolorable Graphs

In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k ≥ 2, a graph G = (V,E) is Pk-bicolorable if its vertex set V can be partitioned into two subsets (i.e., colors) V1 and V2 such that for every induced Pk (i.e., path with exactly k − 1 edges and k vertices) in G, the two colors alternate along the Pk, i.e., no two consecutive vertices of the Pk belong to the same color Vi, i = 1, 2. Obviously, a graph is bipartite if and only if is P2-bicolorable, every graph is Pk-bicolorable for some k and if G is Pk-bicolorable then it is Pk+1-bicolorable. The notion of Pk-bicolorable graphs is motivated by a similar notion of cycle-bicolorable graphs introduced in connection with chordal probe graphs. Moreover, P3and P4-bicolorable graphs are closely related to various other concepts such as 2-subcolorable graphs, P4-bipartite graphs and alternately orientable graphs. We give a structural characterization of P3-bicolorable graphs which also implies linear time recognition of these graphs. Moreover, we give a characterization of P4-bicolorable graphs in terms of forbidden subgraphs.