A LOCAL STRENGTHENING OF REED’S omega, Delta, chi CONJECTURE FOR QUASI- LINE GRAPHS

Abstract

Reed’s omega, Delta, chi conjecture proposes that every graph satisfies chi <= inverted right perpendicular1/2 (Delta + 1 + omega)inverted left perpendicular; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colorings that achieve our bounds: O(n(2)) for line graphs and O(n(3)m(2)) for quasi-line graphs. For line graphs, this is faster than the best known algorithm for constructing a coloring that achieves the bound of Reed’s original conjecture.