Some Ramsey-type numbers and the independence ratio

If each of k, m, and n is a positive integer, there is a smallest positive integer r = rk(m, n) with the property that each graph G with at least r vertices, and with maximum degree not exceeding k, has either a complete subgraph with m vertices, or an independent subgraph with n vertices. In this paper we determine r3(3, n) = r(n), for all n. As a corollary we obtain the largest possible lower bound for the independence ratio of graphs with maximum degree three containing no triangles. From the work of Brooks [2] it follows that if G is a graph with maximum degree k containing no complete graph on k + 1 vertices, then the independence ratio of G is at least \/k. In case G has no complete graph on k vertices, Albertson, Bollobas, and Tucker [1] proved this ratio is larger than \/k, with only two exceptions. And they conjectured that for k = 3, with the additional assumption of planarity, this ratio is bounded away from 1/3. Fajtlowicz [3] verified their conjecture, even without assuming planarity, showing that each cubic graph without triangles has independence ratio at least 12/35. In addition, he displayed a graph in which the independence ratio is exactly 5/14. It follows from our main theorem that 5/14 is a lower bound for the independence ratio in the case k = 3, and in light of Fajtlowicz' graph, 5/14 is the best possible lower bound. In what follows, all graphs will be finite symmetric graphs with no loops and no multiple edges. If G is a graph, then v(G) and e(G) will be the numbers of vertices and edges of G. If M is a set of vertices of G, no two of which are joined by an edge, then M is called independent. The number of vertices in a largest independent vertex set in G will be denoted i(G). A cycle with « vertices will be denoted C„. Proposition I. If G is a graph in which each vertex has degree two or degree three, and if v(G) is odd, then either there is a vertex of degree two both of whose neighbors are of degree two, or else there is a vertex of degree two both of whose neighbors are of degree three. Received by the editors November 6, 1978. AMS (MOS) subject classifications (1970). Primary 05C99.