Tomáš Kaiser

Transversals of d-Intervals

Abstract. We present a method which reduces a family of problems in combinatorial geometry (concerning multiple intervals) to purely combinatorial questions about hypergraphs. The main tool is the Borsuk—Ulam theorem together with one of its extensions. For a positive integer d, a homogeneous d-interval is a union of at most d closed intervals on a fixed line ℓ. Let

Hamilton cycles in 5-connected line graphs

{\cal H}

The Prism Over the Middle-levels Graph is Hamiltonian

be a system of homogeneous d-intervals such that no k + 1 of its members are pairwise disjoint. It has been known that its transversal number

Neighborhood unions and extremal spanning trees

\tau ({\cal H})

Cycles Intersecting Edge-Cuts of Prescribed Sizes

can then be bounded in terms of k and d. Tardos [9] proved that for d = 2, one has

Replication in critical graphs and the persistence of monomial ideals

\tau ({\cal H}) \leq 8k

The rainbow connection number of 2-connected graphs

. In particular, the bound is linear in k. We show that the latter holds for any d, and prove the tight bound

Short Cycle Covers of Graphs with Minimum Degree Three

\tau ({\cal H}) \leq 3k

Contractible subgraphs, Thomassen's conjecture and the dominating cycle conjecture for snarks

for d = 2. We obtain similar results in the case of nonhomogeneous d-intervals whose definition appears below.

Perfect matchings with restricted intersection in cubic graphs

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness.