Mike Newman

On the asymptotic proportion of connected matroids

Very little is known about the asymptotic behavior of classes of matroids. We make a number of conjectures about such behaviors. For example, we conjecture that asymptotically almost every matroid: has a trivial automorphism group; is arbitrarily highly connected; and is not representable over any field. We prove one result: the proportion of labeled n-element matroids that are connected is asymptotically at least 1/2.

On excluded minors for real-representability

We show that for any infinite field K and any K-representable matroid N there is an excluded minor for K-representability that has N as a minor.

On the b-chromatic number of cartesian products

Abstract We consider the b-chromatic number of cartesian products of graphs. We show that the b-chromatic number of K n □ d for d ≥ 3 is one more than the degree; for d ≥ 12 this follows from a result of Kratochvil, Tuza and Voigt. We show that K m □ K n has b-chromatic number at most its degree, and give different approaches that come close to this bound. We also consider cartesian powers of general graphs, and show that the cartesian product of d graphs each with b-chromatic number n is at least d ( n − 1 ) + 1 . This extends a theorem of Kouider and Maheo by removing their condition on independent sets as long as the factor graphs all have the same b-chromatic number.