Sharp bounds for Duke--Erdős--Rödl problems on cycle-connected subgraphs

Abstract

For $k\ge2$, a graph is $C_{2k}$-connected if every two distinct edges lie together on an even cycle of length at most $2k$ in that graph. Let $f_k(n,m)$ be the largest number of edges in a $C_{2k}$-connected subgraph that every $n$-vertex graph with $m$ edges must contain. In this paper, we study two cycle-connected problems raised by Duke, Erdős and Rödl. First, Duke, Erdős and Rödl showed that $f_3(n,\lfloor n^{2-β}\rfloor)=Θ(n^{2-3β})$ for every fixed $0< β<1/2$ and ask for the case of $β\ge 1/2$. We solve this problem in a stronger form by determining that the optimal range of $β$ is $(0,2/3]$. Second, we prove that $f_2(n,\binom n2-s)=(1/(2e)+o(1))n^3/s$ whenever $\frac{s}{n}\to\infty$ and $s=o(n^2)$, determining the leading constant in an estimate of Duke, Erdős, and Rödl. The first result uses a finite-field construction and a count of three-edge paths. The second uses the known structure of $C_4$-connected graphs and estimates for complete bipartite subgraphs in a random graph.