Short disjoint cycles in cubic bridgeless graphs

Abstract

The girth g(G) of a finite simple undirected graph G = (V, E) is defined as the minimum length of a cycle in G. We develope a technique which shows the existence of Ω(n1/7) pairwise disjoint cycles of length 0(n6/7) in cubic bridgeless graphs. As a consequence, for bridgeless graphs with deg v ε {2,3} for all v ε V and {v: deg v = 3}|/|{v: deg v = 2} |≥ c > 0 the girth g(G) is bounded by 0(n6/7). Furthermore similarly as for cycles, the existence of many small disjoint subgraphs with k vertices and k + 2 edges is shown. This very technical result is useful in solving the bisection problem (configuring transputer networks) for regular graphs of degree 4 as B. Monien pointed out. Furthermore the existence of many disjoint cycles in such graphs could be also of selfstanding interest.