Cycle-connected graphs

Abstract

Let X be a fixed collection of graphs. A graph G is .X-connected if every pair of edges of G are contained in a subgraph K of G, where K is a member of 3%. So, for example, if X consists of all paths, then, ignoring isolated vertices, X-connectedness is equivalent to connectedness. If X consists of all paths of length at most d, then each X-connected graph has diameter at most d, while a graph of diameter d is X-connected for X the collection of all paths of length d + 2. In [2,3] we have considered various external problems dealing with the number of edges a graph must have to insure that it contains a large X-connected subgraph when .‘X consists entirely of cycles. The answer may not be trivial even when X consists of only a small number of graphs. For example, suppose X consists of just two cycles, one of length 4 and the other of length 6. In this case we have shown that there exists a positive constant c such that if G is a graph with n vertices and m = dn2 edges, where d = d(n) is a function of n with d(n) 2 n-f, then G must contain a X-connected subgraph with at least cd’n2 = cmW4 edges. Except for the value of c this value is the best possible. If .X consists of all even-length cycles of length at most 12 we have that a graph with m = dn2 edges, d as before, must contain a X-connected subgraph with cd2n2 = cm2np2 edges. Here cm2np2 is also best possible since our graph could be the union of n’m-’ complete bipartite subgraphs, each with cm2np2 edges. It may be true that each graph with m = dn2 edges will still contain a X-connected subgraph with cm2ne2