Ramsey Numbers for Line Graphs and Perfect Graphs

For any graph class \({\cal G}\) and any two positive integers i and j, the Ramsey number \(R_{\cal G}(i,j)\) is the smallest integer such that every graph in \({\cal G}\) on at least \(R_{\cal G}(i,j)\) vertices has a clique of size i or an independent set of size j. For the class of all graphs Ramsey numbers are notoriously hard to determine, and the exact values are known only for very small integers i and j. For planar graphs all Ramsey numbers can be determined by an exact formula, whereas for claw-free graphs there exist Ramsey numbers that are as difficult to determine as for arbitrary graphs. No further graph classes seem to have been studied for this purpose. Here, we give exact formulas for determining all Ramsey numbers for various classes of graphs. Our main result is an exact formula for all Ramsey numbers for line graphs, which form a large subclass of claw-free graphs and are not perfect. We obtain this by proving a general result of independent interest: an upper bound on the number of edges any graph can have if it has bounded degree and bounded matching size. As complementary results, we determine all Ramsey numbers for perfect graphs and for several subclasses of perfect graphs.