clique are equal, and in which this equality persists in every induced subgraph. It is now known (the strong perfect graph theorem) that perfect graphs may be characterized as the graphs that do not have as induced subgraphs either an odd cycle or the complement of an odd cycle (a so-called odd hole). However, for many years this remained an unsolved conjecture, only proven for special subclasses of graphs. One of these subclasses was the family of claw-free graphs: it was discovered by several authors that claw-free graphs without odd cycles and odd holes are perfect. Perfect claw-free graphs may be recognized in polynomial time. In a perfect claw-free graph, the neighborhood of any vertex forms the complement of a bipartite graph. It is possible to color perfect claw-free graphs, or to find maximum cliques in them, in polynomial time. In general, it is NP-hard to find the largest clique in a claw-free graph. It is also NP-hard to find an optimal coloring of the graph, because (via line graphs) this problem generalizes the NP-hard problem of computing the chromatic index of a graph. For the same reason, it is NP-hard to find a coloring that achieves an approximation ratio better than 4/3. However, an approximation ratio of two can be achieved by a greedy coloring algorithm, because the chromatic number of a claw-free graph is greater than half its maximum degree. A generalization of the edge list coloring conjecture states that, for claw-free graphs, the list chromatic number equals the chromatic number; these two numbers can be far apart in other kinds of graphs. The claw-free graphs are χ-bounded, meaning that every claw-free graph of large chromatic number contains a large clique. More strongly, it follows from Ramsey's theorem that every claw-free graph of large maximum degree contains a large clique, of size roughly proportional to the square root of the degree. For connected claw-free graphs that include at least one three-vertex independent set, a stronger relation between chromatic number and clique size is possible: in these graphs, there exists a clique of size at least half the chromatic number. Although not every claw-free graph is perfect, claw-free graphs satisfy another property, related to perfection. A graph is called domination perfect if it has a minimum dominating set that is independent, and if the same property holds in all of its induced subgraphs. Claw-free graphs have this property. To