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arXiv · 2611.07421

Prescribed Color Patterns in Tight Hamilton Cycles and Spanning Trees

Lirong Deng
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Abstract

We compare transversal and pattern thresholds for two spanning structures. For every fixed $k\ge 3$ and $α>0$, we prove that every sufficiently large $n$-vertex $k$-graph collection with minimum codegree at least $(1/2+α)n$ contains a tight Hamilton cycle with any prescribed color pattern. Hence the transversal and pattern thresholds for tight Hamilton cycles coincide asymptotically. For spanning trees, we show that for every $α>0$ there exists $c=c(α)>0$ such that every $n$-vertex graph collection with minimum degree at least $(1/2+α)n$ contains every $n$-vertex tree with maximum degree at most $n^c$ and any prescribed color pattern. On the other hand, we construct spanning trees of maximum degree $\sqrt n+1$ whose pattern threshold exceeds their transversal threshold by a positive linear term.

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Lirong Deng. 2026-11-05. Prescribed Color Patterns in Tight Hamilton Cycles and Spanning Trees. https://arxiv.org/abs/2611.07421

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