A note on cycles in cyclically $4$-edge-connected cubic planar graphs

TL;DR

This paper proves new cycle length bounds in certain cubic planar graphs and their line graphs, using Euler's formula and the Three Edge Lemma, motivated by Bondy and Malkevitch's conjectures.

Contribution

It provides a short proof relating cycle lengths in specific cubic planar graphs to their line graphs, integrating classical lemmas in a novel way.

Findings

If H has circumference at least k, it contains a cycle between k and 3k/2 in length.

The line graph G of Y contains cycles of all lengths avoiding any given vertex.

The proofs combine Euler's formula and the Three Edge Lemma in a new approach.

Abstract

Let $H$ be obtained from a cyclically $4$-edge-connected cubic planar graph $Y$ other than $K_4$ by deleting two adjacent vertices. We provide a short proof that if $H$ has circumference at least $k$ for some even integer $k \ge 4$, then $H$ contains a cycle of length between $k$ and $3k/2$. As a consequence, we show that the line graph $G$ of $Y$ contains a cycle of length $l$ avoiding any prescribed vertex of $G$, for every $l \in \{3\} \cup \{5, \dots, |V(G)| - 1\}$. The proofs integrate Euler's formula and the Three Edge Lemma, established by Thomas and Yu, and independently by Sanders, in a novel way. This work was partially motivated by conjectures of Bondy and Malkevitch.