Bound vertices of longest paths between two vertices in cubic graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917986477015040 |
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| author | Li, Chengli Liu, Feng |
| author_facet | Li, Chengli Liu, Feng |
| contents | Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. This is one of the most important unsolved problems in graph theory. Let $H$ be a subgraph of a graph $G$. A vertex $v$ of $H$ is said to be $H$-bound if all the neighbors of $v$ in $G$ lie in $H$. Recently, Zhan has made the more general conjecture that in a $k$-connected graph, every longest path $P$ between two vertices contains at least $k-1$ internal $P$-bound vertices. In this paper, we prove that Zhan's conjecture holds for $2$-connected cubic graphs. This conclusion generalizes a result of Thomassen [{\em J. Combin. Theory Ser. B} \textbf{129} (2018) 148--157]. Furthermore, we prove that if the two vertices are adjacent, Zhan's conjecture holds for $3$-connected cubic graphs, from which we deduce that every longest cycle in a $3$-connected cubic graph has at least two chords. This strengthens a result of Thomassen [{\em J. Combin. Theory Ser. B} \textbf{71} (1997) 211--214]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bound vertices of longest paths between two vertices in cubic graphs Li, Chengli Liu, Feng Combinatorics 05C38, 05C35, 05C40 Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. This is one of the most important unsolved problems in graph theory. Let $H$ be a subgraph of a graph $G$. A vertex $v$ of $H$ is said to be $H$-bound if all the neighbors of $v$ in $G$ lie in $H$. Recently, Zhan has made the more general conjecture that in a $k$-connected graph, every longest path $P$ between two vertices contains at least $k-1$ internal $P$-bound vertices. In this paper, we prove that Zhan's conjecture holds for $2$-connected cubic graphs. This conclusion generalizes a result of Thomassen [{\em J. Combin. Theory Ser. B} \textbf{129} (2018) 148--157]. Furthermore, we prove that if the two vertices are adjacent, Zhan's conjecture holds for $3$-connected cubic graphs, from which we deduce that every longest cycle in a $3$-connected cubic graph has at least two chords. This strengthens a result of Thomassen [{\em J. Combin. Theory Ser. B} \textbf{71} (1997) 211--214]. |
| title | Bound vertices of longest paths between two vertices in cubic graphs |
| topic | Combinatorics 05C38, 05C35, 05C40 |
| url | https://arxiv.org/abs/2406.07942 |