On the connectivity of p-diamond-free graphs

Let G be a graph of order n(G), minimum degree (G) and connectivity (G). Chartrand and Harary [Graphs with prescribed connectivities, in: P. Erdös, G. Katona (Eds.), Theory of Graphs, Academic Press, NewYork, 1968, pp. 61–63] gave the following lower bound on the connectivity (G) 2 (G)+ 2− n(G). Topp and Volkmann [Sufficient conditions for equality of connectivity and minimum degree of a graph, J. Graph Theory 17 (1993) 695–700] improved this bound to (G) 4 (G)− n(G), if G is bipartite and (G)< (G). In this paper, we show that this result remains valid for diamond-free graphs with (G) 3. A diamond is a graph obtained from a complete graph with 4 vertices by removing an arbitrary edge. Furthermore, we show that the above bounds can be improved significantly for graphs with (G) 3 and no 4-cycle, namely, if (G)< (G), then (G) 2 (G)2 + 2− 2 (G)− n(G). For graphs that, in addition, contain no 3-cycle, we improve this bound even further. © 2007 Published by Elsevier B.V.