Cycles in Color-Critical Graphs
Benjamin Moore, Douglas B. West

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 modulo is -colorable. We prove that if a graph has an edge such that is -colorable and is not, then for , the edge lies in at least cycles of length in , and contains at least cycles of length . A -coloring of is a homomorphism from to the graph with vertex set defined by making and adjacent if . When and are relatively prime, define by . A result of Zhu [2002] implies that is -colorable when has no cycle with length congruent to modulo for any . In fact, only classes need be excluded: we prove that if is…
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