On a problem of Duke-Erdos-Rodl on cycle-connected subgraphs

TL;DR

This paper proves a tight bound on the existence of cycle-connected subgraphs in graphs with a specific number of edges, settling a conjecture and revealing structural properties related to cycle lengths.

Contribution

It establishes the optimal bounds for cycle-connected subgraphs in graphs with many edges, confirming a conjecture by Duke, Erdos, and Rodl.

Findings

Existence of cycle-connected subgraphs with many edges for eta<1/5

Cycle length bounds for edges sharing a vertex

Result is optimal up to constant factors

Abstract

In this short note, we prove that for \beta < 1/5 every graph G with n vertices and n^{2-\beta} edges contains a subgraph G' with at least cn^{2-2\beta} edges such that every pair of edges in G' lie together on a cycle of length at most 8. Moreover edges in G' which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdos, and Rodl.