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Universality for transversal powers of Hamilton cycles

Emily Heath, Joseph Hyde, Natasha Morrison, Shannon Ogden

TL;DR

This work proves that for a collection ${\bf G}={G_1,\dots,G_m}$ of graphs on a common vertex set with minimum degree $\delta({\bf G})\ge\bigl(1-\frac{1}{2k}+\alpha\bigr)n$, every edge-colouring of the $k$th power of a Hamilton cycle $C_n^k$ admits a $\chi$-coloured copy located in ${\bf G}$. The proof adapts an absorption framework to the $k$-th power setting, constructing a reservoir and a robustly absorptive structure built from small gadgets, and an almost spanning collection of $k$-paths via a nibble-style random process. A key contribution is the introduction of absorbing gadgets $F^k_{\ell}$ with low degeneracy and a robust template to assemble them, enabling a final colouring-compatible Hamilton $k$-cycle to be completed. The results generalize previous transversal-embedding theorems (BMPS) to powers of Hamilton cycles and provide insights into the interplay between Dirac-type degree conditions and pattern-specific transversals, while also outlining near-optimality limits and open questions on potential refinements for particular colour patterns.

Abstract

Let $k \ge 2$ and let $\bf G = \{G_1, \ldots, G_{m}\}$ be a collection of graphs on a common vertex set of cardinality $n$. We show that if each graph in $\bf G$ has minimum degree at least $(1-\frac{1}{2k} + o(1))n$, then for every edge-colouring $χ$ of the $k$th power of a Hamilton cycle $C_n^k$ with $m$ colours, there is a copy of $C_n^k$ in $\bf G$ such that $e \in G_{χ(e)}$ for every edge $e$ in $C_n^k$. This generalises a result of Bowtell, Morris, Pehova, and Staden, who provided asymptotically best possible minimum degree conditions for the Hamilton cycle.

Universality for transversal powers of Hamilton cycles

TL;DR

This work proves that for a collection of graphs on a common vertex set with minimum degree , every edge-colouring of the th power of a Hamilton cycle admits a -coloured copy located in . The proof adapts an absorption framework to the -th power setting, constructing a reservoir and a robustly absorptive structure built from small gadgets, and an almost spanning collection of -paths via a nibble-style random process. A key contribution is the introduction of absorbing gadgets with low degeneracy and a robust template to assemble them, enabling a final colouring-compatible Hamilton -cycle to be completed. The results generalize previous transversal-embedding theorems (BMPS) to powers of Hamilton cycles and provide insights into the interplay between Dirac-type degree conditions and pattern-specific transversals, while also outlining near-optimality limits and open questions on potential refinements for particular colour patterns.

Abstract

Let and let be a collection of graphs on a common vertex set of cardinality . We show that if each graph in has minimum degree at least , then for every edge-colouring of the th power of a Hamilton cycle with colours, there is a copy of in such that for every edge in . This generalises a result of Bowtell, Morris, Pehova, and Staden, who provided asymptotically best possible minimum degree conditions for the Hamilton cycle.
Paper Structure (7 sections, 16 theorems, 9 equations, 3 figures, 1 algorithm)

This paper contains 7 sections, 16 theorems, 9 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main lemmas and proof overview
  3. Preliminaries
  4. Main result
  5. Absorption
  6. Almost spanning $k$-path collection
  7. Conclusion

Key Result

Theorem 1.2

Let $k\ge 2$. For every $\alpha>0$ there exists $n_0 := n_0(\alpha,k)$ such that for every $n \geq n_0$ the following holds. Let ${\bf G} = \{G_1, \ldots, G_{m}\}$ be a collection of graphs on common vertex set $V$ with $|V| = n$ such that $\delta({\bf G}) \geq (1-\frac{1}{2k} + \alpha)n$. Then, for

Figures (3)

  • Figure 1: A $2$-connector between two pairs of vertices (left) and a $3$-connector from three vertices to a single vertex (right).
  • Figure 2: The absorbing gadget $F^2_4$ (top), where the black edges all have distinct colours not used elsewhere in the absorber, and how it absorbs $a_1$ (bottom left) and $a_2$ (bottom right). The vertices $a_3$ and $a_4$ are absorbed symmetrically.
  • Figure 3: Graphs $G_1$ (red) and $G_2$ (blue), each with minimum degree $\frac{2}{3}n=2p$, and a colour pattern $\chi$ such that ${\bf G}=\{G_1,G_2\}$ does not contain a $\chi$-coloured Hamilton $2$-cycle. In ${\bf G}$, a solid colour partite set represents a copy of $K_p$, a solid edge represents a copy of $K_{p,p}$, and a dotted edge represents a perfect matching.

Theorems & Definitions (23)

  • Theorem 1.2
  • Lemma 2.0: Absorbing $k$-path
  • Lemma 2.0: $k$-path collection
  • Lemma 3.1: Chernoff bounds, FK
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 4.1
  • proof : Proof of \ref{['thm:main Gen']}:
  • ...and 13 more