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Stability of transversal Hamilton cycles and paths

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arxiv 2403.09913 v1 pith:EKKSSISU submitted 2024-03-14 math.CO

classification math.CO
keywords hamiltontransversalcyclesemphgraphldotsmethodpaths
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abstract

Given graphs $G_1,\ldots,G_s$ all on a common vertex set and a graph $H$ with $e(H) = s$, a copy of $H$ is \emph{transversal} or \emph{rainbow} if it contains one edge from each $G_i$. We establish a stability result for transversal Hamilton cycles: the minimum degree required to guarantee a transversal Hamilton cycle can be lowered as long as the graph collection $G_1,\ldots,G_n$ is far in edit distance from several extremal cases. We obtain an analogous result for Hamilton paths. The proof is a combination of our newly developed regularity-blow-up method for transversals, along with the absorption method.

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  1. The semi-inducibility problem

    math.CO 2025-01 accept novelty 7.0 of 10

    The authors determine sharp or almost sharp maximum densities for alternating walks and cycles and for every 4-cycle colour pattern in red-blue complete graphs, and exhibit a positive-coefficient quantum graph whose o...

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