Yiming Liu

Exact bounds for rank-three matroid basis densities

We determine the maximum asymptotic basis density of rank-three matroids with no U_{3,6} minor: it is 64/81. A uniform gap forces every sufficiently dense simplification to be covered by two lines and one point. This gives weighted stability and the exact finite extremizers for a...

A Fano-plane counterexample to a conjecture on intersecting multipartite hypergraphs

We construct a seven-partite seven-uniform hypergraph with 28 vertices and 22 edges in which every two edges meet in at least two vertices and the ordinary vertex cover number is four. The construction is based on the seven lines of the Fano plane and has a cyclic description by...

Generalized Tuza's conjecture for pair-packing number four

For a finite three-uniform hypergraph, the pair-packing number is the maximum number of edges no two of which share a pair, and the pair-cover number is the minimum number of vertex pairs such that every edge contains a selected pair. We prove that pair-packing number four implie...

The vertex cover number of three-intersecting nine-partite hypergraphs

We prove that every nine-partite nine-uniform hypergraph whose distinct edges intersect in at least three vertices has an ordinary vertex cover of size at most four. Combined with the known lower construction of Bishnoi, Das, Morris and Szabó, this gives Ryser(9,3)=4. The compute...

Sharp exponential edge bounds and flag constructions for multipartite intersecting hypergraphs

We construct strictly t-intersecting r-partite r-uniform hypergraphs with fixed ratio r/t = 3 and arbitrarily large vertex cover number, answering Problem 1 of Bishnoi, Das, Morris and Szabo. The construction attains a sharp universal edge bound: every finite, nonempty, simple hy...

Generalized Tuza's conjecture for pair-packing number four

For a finite three-uniform hypergraph, the pair-packing number is the maximum number of edges no two of which share a pair, and the pair-cover number is the minimum number of vertex pairs such that every edge contains a selected pair. We prove that pair-packing number four implie...

The vertex cover number of three-intersecting nine-partite hypergraphs

We prove that every nine-partite nine-uniform hypergraph whose distinct edges intersect in at least three vertices has an ordinary vertex cover of size at most four. Combined with the known lower construction of Bishnoi, Das, Morris and Szabó, this gives Ryser(9,3)=4. The compute...

A Fano-plane counterexample to a conjecture on intersecting multipartite hypergraphs

We construct a seven-partite seven-uniform hypergraph with 28 vertices and 22 edges in which every two edges meet in at least two vertices and the ordinary vertex cover number is four. The construction is based on the seven lines of the Fano plane and has a cyclic description by...

A Fano-plane counterexample to a conjecture on intersecting multipartite hypergraphs

We construct a seven-partite seven-uniform hypergraph with 28 vertices and 22 edges in which every two edges meet in at least two vertices and the ordinary vertex cover number is four. The construction is based on the seven lines of the Fano plane and has a cyclic description by...

The exact pair-cover number of two-intersecting five-uniform families

We prove that every finite family of five-element sets whose distinct members intersect in at least two elements can be covered by at most six two-element sets, each required to be wholly contained in the member it covers. The known eleven-block cyclic example makes the bound sha...

A ten-pair covering theorem for two-intersecting six-uniform families

A pair cover of a family of sets is a collection of two-element sets such that each member of the family contains a whole selected pair. We prove that every two-intersecting family of six-element sets, on an arbitrary ground set, has a cover by at most ten pairs. The general uppe...

The exact pair-cover number of two-intersecting five-uniform families

We prove that every finite family of five-element sets whose distinct members intersect in at least two elements can be covered by at most six two-element sets, each required to be wholly contained in the member it covers. The known eleven-block cyclic example makes the bound sha...

Forest certificates and Hamilton cycles in Barnette graphs with at most six uncovered vertices

For finite simple 3-connected cubic bipartite plane graphs, pairwise vertex-disjoint facial 4-cycles leaving at most six original vertices uncovered force a partition of the faces of length at least six into two induced forests satisfying Florek's colour constraints. Every intern...

Forest certificates and Hamilton cycles in Barnette graphs with at most six uncovered vertices

Barnette's conjecture asks whether every finite 3-connected cubic bipartite planar graph has a Hamiltonian cycle. We show that, for a finite simple 3-connected cubic bipartite plane graph, pairwise vertex-disjoint facial 4-cycles leaving at most six vertices uncovered force a col...

Hamiltonicity of Barnette graphs with six vertices uncovered by disjoint facial 4-cycles

Barnette's conjecture asks whether every finite 3-connected cubic bipartite planar graph has a Hamiltonian cycle. This preprint studies a sufficient condition for a fixed plane embedding. Let G be a finite simple 3-connected cubic bipartite plane graph, and let Q be a family of p...