Qinfei Tang | Search arXiv

arXiv2026

A sharp density bound for 5-connected graphs with no $\Ke$ minor

Let $\Ke$ be obtained from $K_7$ by deleting two independent edges. We prove that every 5-connected graph on $n\ge7$ vertices with at least $4n-9$ edges contains a $\Ke$ minor, settling Conjecture~1.4 of Dvo\v rák, Norin and Rahman (arXiv preprint 2609.17760v1). The bound is sharp. We prove the stronger statement that every $4$-bilight graph on $n\ge4$ vertices with at least $4n-9$ edges contains either a $\Ke$ minor or a $K_6$ subgraph. Within their reduction framework, we strengthen the rooted-minor theorem. We show that every $4$-light 5-rooted graph of rooted $4$-density at least two has a model with two nonroot vertices and at most one missing edge incident with them. At the critical density, reductions preserve density exactly, which prevents them from creating a new $K_6$ subgraph.

math.CO↗

arXiv2026

Layer barriers for colour-biased tight Hamilton cycles

We construct a family of layer barriers for colour-biased tight Hamilton cycles in uniform hypergraphs. For every $k\ge 3$ and every $a\in\{0,\ldots,k-1\}$, 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.

math.CO↗