Cycles in Color-Critical Graphs

TL;DR

This paper investigates the relationship between cycles of specific lengths and colorability in graphs, establishing lower bounds on cycles containing certain edges and extending known coloring results to more general modular conditions.

Contribution

It generalizes Tuza's and Zhu's results by providing new bounds on cycle counts related to color-critical edges and introduces refined conditions for $(k,d)$-colorability based on cycle lengths.

Findings

Edges critical for colorability lie in many cycles of specific lengths.

Graphs with certain cycle restrictions are guaranteed to have multiple cycles of particular lengths.

New bounds improve understanding of cycle structures in color-critical graphs.

Abstract

Tuza [1992] proved that a graph with no cycles of length congruent to $1$ modulo $k$ is $k$-colorable. We prove that if a graph $G$ has an edge $e$ such that $G-e$ is $k$-colorable and $G$ is not, then for $2\leq r\leq k$, the edge $e$ lies in at least $\prod_{i=1}^{r-1}(k-i)$ cycles of length $1\mod r$ in $G$, and $G-e$ contains at least $\frac{1}{2}\prod_{i=1}^{r-1}(k-i)$ cycles of length $0 \mod r$. A $(k,d)$-coloring of $G$ is a homomorphism from $G$ to the graph $K_{k:d}$ with vertex set $\mathbb{Z}_{k}$ defined by making $i$ and $j$ adjacent if $d\leq j-i \leq k-d$. When $k$ and $d$ are relatively prime, define $s$ by $sd\equiv 1\mod k$. A result of Zhu [2002] implies that $G$ is $(k,d)$-colorable when $G$ has no cycle $C$ with length congruent to $is$ modulo $k$ for any $i\in \{1,\ldots,2d-1\}$. In fact, only $d$ classes need be excluded: we prove that if $G-e$ is $(k,d)$-colorable and $G$ is not, then $e$ lies in at least one cycle with length congruent to $is\mod k$ for some $i$ in $\{1,\ldots,d\}$. Furthermore, if this does not occur with $i\in\{1,\ldots,d-1\}$, then $e$ lies in at least two cycles with length $1\mod k$ and $G-e$ contains a cycle of length $0 \mod k$.