Short disjoint cycles in graphs with degree constraints

Abstract

We show that each finite undirected graph G=(V, E), ¦V¦ = n, ¦E¦ = m with minimum degree δ(G)≥ 3 and maximum degree Δ = Δ(G) contains at least n/4(Δ−1)log2n pairwise vertex-disjoint cycles of length at most 4(Δ −1) · log2n. Furthermore collections of such cycles can be determined within O(n · (n + m)) steps. For constant Δ this means Ω(n/logn) cycles of length O(logn). This bound is also an optimum.

A similar approach yields similar bounds for subgraphs with more edges than vertices instead of cycles. Furthermore also collections of many small pairwise disjoint induced subgraphs of this type can be determined within O(n · (n + m)) steps similarly as for cycles.