Let G = (V, E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,. .. , k}) to each vertex of G so that no edge has both endpoints with the same color. We present a new local search algorithm, called Variable Space Search (VSS), which we apply to the k-coloring problem. VSS extends the Formulation Space Search (FSS) methodology by considering several non equivalent formulations of a same problem, each one being associated with a set of neighborhoods and an objective function. The search moves from one formulation to another when it is blocked at a local optimum with a given formulation. The k-coloring problem is thus solved by combining different formulations of the problem which are not equivalent, in the sense that some constraints are possibly relaxed in one search space and always satisfied in another. We show that the proposed algorithm improves on every local search used independently (i.e., with a unique search space), and is competitive with the currently best coloring methods, which are complex hybrid evolutionary algorithms.

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Reed's ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ 1 2 (∆ + 1 + ω) ; 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 colourings 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 colouring that achieves the bound of Reed's original conjecture.

In 1988, Chvátal and Sbihi (J Combin Theory Ser B 44(2) (1988), 154–176) proved a decomposition theorem for claw‐free perfect graphs. They showed that claw‐free perfect graphs either have a clique‐cutset or come from two basic classes of graphs called elementary and peculiar graphs. In 1999, Maffray and Reed (J Combin Theory Ser B 75(1) (1999), 134–156) successfully described how elementary graphs can be built using line‐graphs of bipartite graphs using local augmentation. However, gluing two claw‐free perfect graphs on a clique does not necessarily produce claw‐free graphs. In this article, we give a complete structural description of claw‐free perfect graphs. We also give a construction for all perfect circular interval graphs.

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