https://hal.inria.fr/hal-00958951Hoàng, Chinh T.Chinh T.HoàngDepartment of Physics and Computer Science [Waterloo] - WLU - Wilfrid Laurier UniversityLe, van Bangvan BangLeInstitut für Informatik [Rostock] - Universität RostockP_4-Colorings and P_4-Bipartite GraphsHAL CCSD2001Perfect graphthe Strong Perfect Graph Conjectruegraph partitioncographNP-completeness[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Inria Sophia Antipolis-Méditerranée / I3s, Service Ist2014-03-13 16:51:532018-11-23 15:38:022014-03-13 21:14:05enJournal articleshttps://hal.inria.fr/hal-00958951/document10.46298/dmtcs.272application/pdf1A vertex partition of a graph into disjoint subsets V_is is said to be a P_4-free coloring if each color class V_i induces a subgraph without chordless path on four vertices (denoted by P_4). Examples of P_4-free 2-colorable graphs (also called P_4-bipartite graphs) include parity graphs and graphs with ''few'' P_4s like P_4-reducible and P_4-sparse graphs. We prove that, given k≥ 2, \emphP_4-Free k-Colorability is NP-complete even for comparability graphs, and for P_5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P_4-bipartite graphs with special P_4-structure. In particular, we show that the SPGC is true for P_4-bipartite graphs with one P_3-free color class meeting every P_4 at a midpoint.