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

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 integral weight function c on the vertices of G, the weighted chromatic number of (G, c) can be obtained by rounding up its fractional relaxation. As a corollary, we obtain that the weighted chromatic number can be computed in polynomial-time for these graphs. Finally, we show a new example of a graph operation which preserves the integer round-up property for the chromatic number, and use it to provide a first example of a t-perfect 3-colorable graph which does not have this property.