Girth in graphs

AbstractIt 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)