A tournament is a complete graph with its edges directed, and colouring a tournament means partitioning its vertex set into transitive subtournaments. For some tournaments H there exists c such that every tournament not containing H as a subtournament has chromatic number at most c (we call such a tournament H a hero); for instance, all tournaments with at most four vertices are heroes. In this paper we explicitly describe all heroes.

We present a new efficient method to find the Ising problem partition function for finite lattice graphs embeddable on an arbitrary orientable surface, with integral coupling constants bounded in the absolute value by a polynomial of the size of the lattice graph. The algorithm has been implemented for toroidal lattices using modular arithmetic and the generalized nested dissection method. The implementation has substantially better performance than any other as far as we know.

Kasteleyn stated that the generating function of the perfect matchings of a graph of genus $g$ may be written as a linear combination of $4^g$ Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix.

No abstract

We consider the length L of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k-ary alphabet. Subadditivity arguments yield that E [L] /n converges to a constant γ k . We prove a conjecture of Sankoff and Mainville from the early 80's claiming that γ k √ k → 2 as k → ∞.