*P4*-Colorings and *P4*-Bipartite Graphs

*Chính T. Hoàng, Van Bang Le*

#### Abstract

A vertex partition of a graph into disjoint subsets Vis is said to be a P4-free coloring if each color class Vi induces a subgraph without chordless path on four vertices (denoted by P4). Examples of P4-free 2-colorable graphs (also called P4-bipartite graphs) include parity graphs and graphs with ``few'' P4s like P4-reducible and P4-sparse graphs. We prove that, given k≥2,

*P4-Free k-Colorability*is NP-complete even for comparability graphs, and for P5-free graphs. We then discuss the recognition, perfection and the Strong Perfect Graph Conjecture (SPGC) for P4-bipartite graphs with special P4-structure. In particular, we show that the SPGC is true for P4-bipartite graphs with one P3-free color class meeting every P4 at a midpoint.Full Text: GZIP Compressed PostScript PostScript PDF original HTML abstract page