Chords of longest cycles passing through a specified small set

Abstract

A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is $2$-connected and cubic, then any longest cycle must have a chord. He also showed that if $G$ is a $3$-connected graph with minimum degree at least $4$, then some of the longest cycles in $G$ must have a chord. Zhang (1987) proved that if $G$ is a $3$-connected simple planar graph which is 3-regular or has minimum degree at least $4$, then every longest cycle of $G$ must have a chord. Recently, Li and Liu showed that if $G$ is a $2$-connected cubic graph and $x, y$ are two distinct vertices of $G$, then every longest $(x,y)$-path of $G$ contains at least one internal vertex whose neighbors are all in the path. In this paper, we study chords of longest cycles passing through a specified small set and generalize Thomassen's and Zhang's above results by proving the following results. (i) Let $G$ be a $2$-connected cubic graph and $S$ be a specified set consisting of an edge plus a vertex. Then every longest cycle of $G$ containing $S$ must have a chord. (ii) Let $G$ be a $3$-connected graph with minimum degree at least $4$ and $e$ be a specified edge of $G$. Then some longest cycle of $G$ containing $e$ must have a chord. (iii) Let $G$ be a $3$-connected planar graph with minimum degree at least $4$. Suppose $S$ is a specified set consisting of either three vertices or an edge plus a vertex. Then every longest cycle of $G$ containing $S$ must have a chord. We also extend the above-mentioned result of Li and Liu for $2$-connected cubic graphs.