Dean's conjecture and cycles modulo k

TL;DR

This paper proves Dean's conjecture for all k ≥ 6, showing that graphs with minimum degree at least k contain cycles of all lengths modulo k, except for certain exceptional graphs, and introduces new graph families for analysis.

Contribution

The paper resolves Dean's conjecture for all k ≥ 6 and introduces trigonal and tetragonal graph families to study cycle lengths modulo k.

Findings

Graphs with minimum degree ≥ k contain cycles of all lengths mod k for k ≥ 6.

Exceptional graphs only fail to contain cycles of length 2 mod k.

New graph families aid in analyzing path and cycle lengths.

Abstract

Dean conjectured three decades ago that every graph with minimum degree at least $k\ge 3$ contains a cycle whose length is divisible by $k$. While the conjecture has been verified for $k\in \{3,4\}$, it remains open for $k\ge 5$. A weaker version, also proposed by Dean, asserting that every $k$-connected graph contains a cycle of length divisible by $k$, was resolved by Gao, Huo, Liu, and Ma using the notion of admissible cycles. In this paper, we resolve Dean's conjecture for all $k\ge 6$. In fact, we prove a stronger result by showing that every graph with minimum degree at least $k$ contains cycles of length $r \pmod k$ for every even integer $r$, unless every end-block belongs to a specific family of exceptional graphs, which fail only to contain cycles of length $2 \pmod k$. We also establish a strengthened result on the existence of admissible cycles. Our proof introduces two sparse graph families, called trigonal graphs and tetragonal graphs, which provide a flexible framework for studying path and cycle lengths and may be of independent interest.