Layer barriers for colour-biased tight Hamilton cycles

Abstract

We construct a family of layer barriers for colour-biased tight Hamilton cycles in uniform hypergraphs. For every $k\ge 3$ and every $a\in \{0,\dots ,k-1\}$, we give a red--blue coloured $k$-graph that contains a tight Hamilton cycle, while every tight Hamilton cycle in the construction is perfectly colour-balanced. The construction underlying the higher-uniformity threshold conjectured by Behague, Clemen, Hyde and Morrison corresponds to the boundary case $a=0$ of this family. We show that interior choices of $a$ can yield strictly denser barriers. In particular, for $k=17$ and $a=8$, the asymptotic relative minimum vertex degree of our construction is \[ \frac{5761}{8192}\approx 0.703247, \] which exceeds the conjectured value $d_{17}\approx 0.699277$. This provides a counterexample to the proposed higher-uniformity threshold in Conjecture~6.1 of Behague, Clemen, Hyde and Morrison. Moreover, by choosing the layer appropriately as $k\to \infty$, the family contains barriers whose asymptotic relative minimum vertex degree is \[ 1-O\bigl(k^{-1/2}\bigr). \] Thus the interior members of the layer-barrier family exhibit substantially different behaviour from the previously considered boundary construction in large uniformity.

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