Search algorithm for Ramsey graphs by union of group orbits

Abstract

AbstractAn algorithm for the construction of Ramsey graphs with a given automorphism group G is presented. To find a graph on n vertices with no clique of k vertices, Kk, and no independent set of l vertices, ¯Kl, k, l ≤ n, with an automorphism group G, a Boolean formula α based on the G‐orbits of k‐subsets and l‐subsets of vertices is constructed from incidence matrices belonging to G. This Boolean formula is satisfiable if and only if the desired graph exists, and each satisfying assignment to α specifies a set of orbits of pairs of vertices whose union gives the edges of such a graph. Finding these assignments is basically equivalent to the conversion of α from CNF to DNF (conjunctive to disjunctive normal form). Though the latter problem is NP‐hard, we present an “efficient” method to do the conversion for the formulas that appear in this particular problem. When G is taken to be the dihedral group Dn for n ≤ 101, this method matches all of the previously known cyclic Ramsey graphs, as reported by F. R. K. Chung and C. M. Grinstead [“A Survey of Bounds for Classical Ramsey Numbers,” Journal of Graph Theory, 7 (1983), 25–38], in dramatically smaller computer time when compared to the time required by an exhaustive search. Five new lower bounds for the classical Ramsey numbers are established: R(4, 7) ⩾ 47, R(4, 8) ⩾ 52, R(4, 9) ⩾ 69, R(5,7) ⩾ 76, and R(5, 8) ⩾ 94. Also, some previously known cyclic graphs are shown to be unique up to isomorphism.