Chords of longest cycles passing through a specified small set

TL;DR

The paper advances the study of Thomassen's chord conjecture by focusing on longest cycles that pass through fixed small sets in several graph classes. It provides three new results: in a $2$-connected cubic graphs, in a $3$-connected graphs with $ olinebreak[4] olinebreak[4] olinebreak[4]$ , and in a $3$-connected planar graphs, every longest cycle containing a specified small set has a chord, extending prior work of Thomassen, Zhang, and Li–Liu. The methods combine parity arguments, the lollipop construction, Tutte-type cycle augmentations, and planar-case separability analysis to derive chord existence. It also proposes a broad conjecture for longest cycles through linear forests in $k$-connected graphs and discusses related matroid-theoretic notions and counterexamples.

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.

Theorems & Definitions (17)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7
• Corollary 1.8
• Corollary 1.9
• Theorem 1.10