Bound vertices of longest paths between two vertices in cubic graphs

Abstract

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].

Figures (5)

• Figure 1: The construction of $G_2$
• Figure 2: Illustration of paths corresponding to the maximal blue subpaths
• Figure 3: Illustrations of $C'$ and $C^*$
• Figure 4: Illustration of $G_3$
• Figure 5: Illustrations of $C$ and $C'$

Theorems & Definitions (17)

• Conjecture 1.1: Thomassen Alspach1985Thomassen1989
• Theorem 1.2: Thomassen Thomassen1997
• Conjecture 1.3: Zhan Zhan2024
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• proof : Proof of Corollary \ref{['Cororllary-cubic-3-connected-2-chord']}, assuming Theorem \ref{['Theorem-cubic-3-connected']}
• Theorem 2.1: Fleischner-Stiebitz Fleischner1992
• Lemma 2.2
• proof : Proof of Lemma \ref{['Lemma-color-triangle-path']}.