Girth in graphs

Abstract

It is shown that a graph of large girth and minimum degree at least 3 share many properties with a graph of large minimum degree. For example, it has a contraction containing a large complete graph, it contains a subgraph of large cyclic vertex-connectivity (a property which guarantees, e.g., that many prescribed independent edges are in a common cycle), it contains cycles of all even lengths modulo a prescribed natural number, and it contains many disjoint cycles of the same length. The analogous results for graphs of large minimum degree are due to Mader ( Math. Ann. 194 (1971), 295–312; Abh. Math. Sem. Univ. Hamburg 37 (1972), 86–97), Woodall ( J. Combin. Theory Ser. B 22 (1977), 274–278), Bollobás ( Bull. London Math. Soc. 9 (1977), 97–98) and Häggkvist (Equicardinal disjoint cycles in sparse graphs, to appear). Also, a graph of large girth and minimum degree at least 3 has a cycle with many chords. An analogous result for graphs of chromatic number at least 4 has been announced by Voss ( J. Combin. Theory Ser. B 32 (1982), 264–285).