Lollipops, dense cycles and chords

TL;DR

This paper extends classical results on cycles in graphs with high minimum degree by showing how certain edges can be contracted to produce minors with high minimum degree and large clique minors.

Contribution

It introduces the concept of cyclic minors obtained through edge contractions and establishes conditions for the existence of large clique minors in such cycles.

Findings

Edges can be contracted to produce graphs with high minimum degree

Minimum degree at least O(k^2) guarantees a cyclic K_k-minor

Extends classical cycle and chord results to minors with high connectivity

Abstract

In 1980, Gupta, Kahn and Robertson proved that every graph $G$ with minimum degree at least $k\geq 2$ contains a cycle $C$ containing at least $k+1$ vertices each having at least $k$ neighbors in $C$ (so $C$ has at least $\frac{(k+1)(k-2)}{2}$ chords). In this work, we go further by showing that some of its edges can be contracted to obtain a graph with high minimum degree (we call such a minor of $C$ a \emph{cyclic minor}). We then investigate further cycles having cliques as cyclic minors, and show that minimum degree at least $O(k^2)$ guarantees a cyclic $K_k$-minor.