Existence of cycles of length divisible by 3 or 4

TL;DR

This paper investigates conditions under which graphs with certain minimum degree and degree-2 vertex constraints contain cycles of lengths divisible by 3 or 4, extending and characterizing known conjectures and results.

Contribution

It strengthens existing theorems by characterizing graphs with limited degree-2 vertices that lack cycles divisible by 3 or 4, providing new insights into graph cycle divisibility.

Findings

Graphs with minimum degree at least 2 and at most two degree-2 vertices contain cycles divisible by 3.

Graphs with at least nine vertices, minimum degree at least 2, and at most three degree-2 vertices contain cycles divisible by 4.

Complete characterizations of graphs with up to three degree-2 vertices lacking cycles divisible by 3 or 4.

Abstract

Dean conjectured that for each integer $k \ge 3$, every graph with minimum degree at least $k$ has a cycle whose length is divisible by $k$; this conjecture is known to be true for all $k\neq 5$. For $k\in\{3,4\}$, stronger statements are true: every graph with minimum degree at least $2$ and at most $k-2$ vertices of degree $2$ has a cycle whose length is divisible by $k$. We further strengthen these results by characterizing all graphs with minimum degree at least $2$ and at most three vertices of degree $2$ that have no cycle of length divisible by $k$, for each $k\in\{3,4\}$. As a corollary, we obtain that every graph with minimum degree at least $2$ and at most two vertices of degree $2$ has a cycle whose length is divisible by $3$, and that every graph on at least nine vertices with minimum degree at least $2$ and at most three vertices of degree $2$ has a cycle whose length is divisible by $4$.