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Cycles in Color-Critical Graphs

2021
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Electronic Journal of Combinatorics
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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 $2\le r\le k$, 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 $\frac12{\prod_{i=1}^{r-1} (k-i)}$ 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

doi:10.37236/10177
fatcat:hd7jl2wu5vcjfjklku74f3doba