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Patrick White

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Preprint Sep 2026

Tuza's Ryser-conjecture claim for four-partite hypergraphs with matching number two

We prove that every $4$-partite $4$-uniform hypergraph $H$ with matching number $\nu(H)=2$ satisfies $\tau(H)\le 6$, where $\tau$ denotes the vertex-cover number. This confirms a claim made by Tuza in his 1979 manuscript but never published with a proof, and closes the case $(r,\nu)=(4,2)$ of Ryser's conjecture. The be...

Patrick White · 0 citations

Preprint Jul 2026

New bounds for double covers of the discrete box {0,1,2}^d

A proper sub-box of $A=\{0,1,2\}^d$ is a product $S_1\times\dots\times S_d$ with each $\varnothing\neq S_i\subsetneq\{0,1,2\}$. A double cover is a finite multiset of proper sub-boxes covering every point of $A$ exactly twice; write $f(d)$ for the minimum size of a double cover. Leader, Milicevic and Tan asked whether...

Patrick White · 0 citations

Preprint Jul 2026

The small Davenport constant of the Heisenberg group of order 125

The small Davenport constant $\mathsf{d}(G)$ of a finite group $G$ is the maximal length of a product-one-free sequence over $G$. For the exponent-$p$ Heisenberg group $H_{p^3}$ of order $p^3$, Godara and Sarkar proved $\mathsf{d}(H_{27})=6$ and posed $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$, leaving $p\ge5$...

Patrick White · 2 citations