Integer round-up property for the chromatic number of some h-perfect graphs

A graph is h-perfect if its stable set polytope can be completely described by nonnegativity, clique and odd-hole constraints. It is t-perfect if it furthermore has no clique of size 4. For every graph G and every c ∈ Z (G) + , the weighted chromatic number of (G, c) is the minimum cardinality of a multi-set F of stable sets of G such that every v ∈ V (G) belongs to at least cv members of F . We prove that every h-perfect line-graph and every t-perfect claw-free graph G has the integer round-up property for the chromatic number: for every non-negative integer weight c on the vertices of G, the weighted chromatic number of (G, c) can be obtained by rounding up its fractional relaxation. In other words, the stable set polytope of G has the integer decomposition property. Another occurrence of this property was recently obtained by Eisenbrand and Niemeier for fuzzy circular interval graphs (extending previous results of Niessen, Kind and Gijswijt). These graphs form another proper subclass of claw-free graphs. Our results imply the existence of a polynomial-time algorithm which computes the weighted chromatic number of t-perfect claw-free graphs and h-perfect line-graphs. Finally, they yield a new case of a conjecture of Goldberg and Seymour on edge-colorings.