Dean's conjecture and cycles modulo k

TL;DR

This work resolves Dean's conjecture for all $k \ge 6$ by proving that every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$, except when every end-block lies in a small exceptional family. The authors introduce two sparse core graph families, trigonal (non-bipartite) and tetragonal (bipartite), to generate long arithmetic progressions of path lengths between vertex pairs, enabling concatenation into $k$ admissible cycles. A key reduction places the problem within the framework of $k$-weak graphs (Type I/II), and the main results show these graphs contain cycles of all even residues modulo $k$ unless they are in the exceptional families, yielding the corollary that cycles exist for all even residues except possibly $2 \pmod{k}$ in the obstruction cases. The techniques, including core-sparse structures and admissible cycles, advance the modular-cycle-length theory and may have independent applicability to related cycle-length problems in graphs. The methods provide not only the proof of Dean's conjecture for $k \ge 6$ but also a strengthened existence result for admissible cycles, along with a constructive framework (core subgraphs) that could be useful in broader extremal graph theory problems. The results sharpen the understanding of how minimum-degree conditions translate into prescribed-cycle-length residues, offering new tools for explicit residue control in 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.

Figures (4)

• Figure 1: Trigonal graphs
• Figure 2: All trigonal graphs on at most six vertices
• Figure 3: Tetragonal graphs
• Figure 4: Forming larger tetragonal graphs in Lemma \ref{['lem:operation']}

Theorems & Definitions (77)

• Conjecture 1.1: Dean's conjecture
• Theorem 1.2: Main Theorem
• Corollary 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.3
• Theorem 2.4
• Lemma 2.5
• proof
• Definition 3.1: Trigonal graph