Crux and Long Cycles in Graphs

Abstract

We introduce a notion of the crux of a graph $G$, measuring the order of a smallest dense subgraph in $G$. This simple-looking notion leads to some generalizations of known results about cycles, offering an interesting paradigm of “replacing average degree by crux.” In particular, we prove that every graph contains a cycle of length linear in its crux. Long proved that every subgraph of a hypercube $Q^m$ (resp., discrete torus $C_3^m$) with average degree $d$ contains a path of length $2^{d/2}$ (resp., $2^{d/4}$) and conjectured that there should be a path of length $2^{d}-1$ (resp., $3^{d/2}-1$). As a corollary of our result, together with isoperimetric inequalities, we close these exponential gaps giving asymptotically optimal bounds on long paths in hypercubes, discrete tori, and more generally Hamming graphs. We also consider random subgraphs of $C_4$-free graphs and hypercubes, proving near optimal lower bounds on the lengths of long cycles.