Lollipops, dense cycles and chords

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.

Figures (7)

• Figure 1: Two representations of a lollipop $L$.
• Figure 2: Construction of an element of $\mathcal{S}_{i+1}$ from an element of $\mathcal{S}_i$.
• Figure 3: A situation from the proof of \ref{['lem:kactive']}.
• Figure 4: Another situation from the proof of \ref{['lem:kactive']}.
• Figure 5: A permutation matrix

Theorems & Definitions (26)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5