Roland Häggkvist

On multiplicative graphs and the product conjecture

We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Nešetřil and Pultr These results allow us (in conjunction with earlier results of Nešetřil and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].

CYCLES AND PATHS IN BIPARTITE TOURNAMENTS WITH SPANNING CONFIGURATIONS

We give necessary and sufficient conditions in terms of connectivity and factors for the existence of hamiltonian cycles and hamiltonian paths and also give sufficient conditions in terms of connectivity for the existence of cycles through any two vertices in bipartite tournaments.

Sorting and Merging in Rounds

The need for sorting algorithms which operate in a fixed number of rounds (rather than have each new comparison depend on the outcomes of all previous comparisons) arises in structural modeling. Since all comparisons within a round are evaluated simultaneously, such algorithms have an obvious connection to parallel processing.In an earlier paper (SIAM J. Comput.,10 (1981), pp. 465–472) we used a counting argument to prove the existence of subquadratic sorting algorithms for two rounds. Here we develop optimal algorithms for merging in rounds, and apply them to actually construct good sorting algorithms for k rounds,

Trees in tournaments

k\geqq 3

Powers of Hamilton cycles in tournaments

. For example, in

On numerical semigroups

k = 66

A note on list‐colorings

rounds, our algorithm will sort any n-element linearly ordered set with

Oriented Hamilton Cycles in Oriented Graphs

O ( n^{1.10} )

Dissecting a square into rectangles of equal area

comparisons.

Gorenstein rings as maximal subrings of k[[x]] with fixed conductor

Letf(n) be the smallest integer such that every tournament of orderf(n) contains every oriented tree of ordern. Sumner has just conjectures thatf(n)=2n−2, and F. K. Chung has shown thatf(n)≤(1+o(1))nlog2n. Here we show thatf(n)≤12n andf(n)≤(4+o(1))n.