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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2605.03786 |
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| _version_ | 1866917476465377280 |
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| author | Lo, On-Hei Solomon |
| author_facet | Lo, On-Hei Solomon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03786 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on cycles in cyclically $4$-edge-connected cubic planar graphs Lo, On-Hei Solomon Combinatorics 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. |
| title | A note on cycles in cyclically $4$-edge-connected cubic planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.03786 |