The class of (P7,C4,C5)‐free graphs: Decomposition, algorithms, and χ‐boundedness

Abstract

As usual, P n ( n ≥ 1) denotes the path on n vertices, and C n ( n ≥ 3) denotes the cycle on n vertices. For a family MJX-tex-caligraphicscriptH of graphs, we say that a graph G is MJX-tex-caligraphicscriptH‐free if no induced subgraph of G is isomorphic to any graph in MJX-tex-caligraphicscriptH. We present a decomposition theorem for the class of ( P 7 , C 4 , C 5 )‐free graphs; in fact, we give a complete structural characterization of ( P 7 , C 4 , C 5 )‐free graphs that do not admit a clique‐cutset. We use this decomposition theorem to show that the class of ( P 7 , C 4 , C 5 )‐free graphs is χ‐bounded by a linear function (more precisely, every ( P 7 , C 4 , C 5 )‐free graph G satisfies χ ( G ) ≤ 3 / 2 ω ( G )). We also use the decomposition theorem to construct an O ( n 3 ) algorithm for the minimum coloring problem, an O ( n 2 m ) algorithm for the maximum weight stable set problem, and an O ( n 3 ) algorithm for the maximum weight clique problem for this class, where n denotes the number of vertices and m the number of edges of the input graph.

On coloring of graphs of girth 2l + 1 without longer odd holes

“…Answering a question of Kalai and Meshulam, Bonamy, Charbit and Thomassé [2] proved that there is a constant c such that χ(G) ≤ c for each graph G ∈ H. The question that whether χ(G) ≤ 3 for each G ∈ H remains open. There are also quite a lot of results concerning the structure and chromatic number of graphs inducing no paths on l vertices, one may see [3][4][5]8,10,15,18,19] for some most recent results. Interested readers are referred to [23] and [26] for more information on χ-bounded problems.…”