Stability of transversal Hamilton cycles and paths

• DOI: 10.13140/rg.2.2.18514.77760
• Publication Date: 2024-01-01
• Authors: Cheng, Yangyang; Staden, Katherine
• Keywords: FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO)

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.