arXiv subjects

Qinfei Tang

Publications and source records attributed to Qinfei Tang.

3 recordsLinked to original sources

arXiv2026

Graph and Hypergraph Counterexamples to Mubayi's supersaturation Conjecture Beyond $n^{k-1}$

For a fixed $k$-graph $F$, let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex $k$-graph with $\text{ex}(n,F)+q$ edges. Let $c(n,F)$ be the minimum number created by adding one edge to an extremal $F$-free $k$-graph. We construct stable counterexamples to Mubayi's supersaturation conjecture for every $k\ge2$. The ratio $h_F(n,q)/(q c(n,F))$ tends to zero over ranges extending beyond $n^{k-1}$. For graphs, we give examples of every chromatic number at least three. In particular, there is a graph $F$ for which this ratio tends to zero for $1\le q=o(n^{4/3})$. For $k\ge3$, we use connected linear expansions of $G=(M_a\cup K_{1,b}\cup I_h)\vee K_{m_1,\ldots,m_{k-1}}$. Under explicit parameter conditions, the ratio tends to zero for $1\le q=o(n^{k-1+(b-1)/(a+b-1)})$ when $b\ge k+1$. We prove the exact Turán theorems for both constructions and determine their unique extremal graphs and hypergraphs. Keywords: supersaturation; extremal graph; linear hypergraph; graph expansion; stability.

math.CO↗

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↗