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The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

Perfect graphs were defined by Claude Berge in the 1960s. They are important objects for graph theory, linear programming and combinatorial optimization. Claude Berge made a conjecture about them, that was proved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and is now called the strong perfect graph theorem. This is a survey about perfect graphs, mostly focused on the strong perfect gr...

Object oriented development is the methodology of choice for a wide range of applications but those developing critical systems have stayed with techniques with a mathematical basis. Object-Z is a formal specication language that has attempted to bring the benets of an object oriented approach to critical systems. However, it lacks an obvious route to implementation. This paper describes how Ob...

We define a diamonded odd cycle to be an odd cycle C with exactly two chords and either a) C has length five and the two chords are non-crossing; or b) C has length greater than five and has chords (x,y) and (x,z) with (y,z) an edge of C and there exists a node w not on C adjacent to y and C, but not x. In this paper, we show that given a diamonded odd cycle-free graph G, G is perfect if and on...

In 1979, two constructions for making partitionable graphs were introduced in (by Chv2 atal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A; B) of a %nite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to b...