Note on the game chromatic index of trees

We study edge coloring games defining the so-called game chromatic index of a graph. It has been reported that the game chromatic index of trees with maximum degree $\Delta = 3$ is at most $\Delta + 1$. We show that the same holds true in case $\Delta \geq 6$, which would leave only the cases $\Delta = 4$ and $\Delta = 5$ open. \u

On pseudomodular matroids and adjoints

AbstractThere are two concepts of duality in combinatorial geometry. A set theoretical one, generalizing the structure of two orthocomplementary vector spaces, and a lattice theoretical concept of an adjoint, that mimics duality between points and hyperplanes. The latter — usually called polarity — seems to make sense almost only in the linear case. In fact the only non-linear combinatorial geometries known to admit an adjoint were of rank 3. Moreover, N.E. Mnëv conjectured that in higher ranks there would exist no non-linear oriented matroid that has an oriented adjoint. At least with unoriented matroids this is not true. In this paper we present a class of rank-4 matroids with adjoint including a non-linear example

Using network-flow techniques to solve an optimization problem from surface-physics

The solid-on-solid model provides a commonly used framework for the description of surfaces. In the last years it has been extended in order to investigate the effect of defects in the bulk on the roughness of the surface. The determination of the ground state of this model leads to a combinatorial problem, which is reduced to an uncapacitated, convex minimum-circulation problem. We will show that the successive shortest path algorithm solves the problem in polynomial time.Comment: 8 Pages LaTeX, using Elsevier preprint style (macros included

About the Tic-Tac-Toe Matroid

The purpose of this note is to make a problem, already mentioned in [3], more tangible. We introduce a matroid which has "the" combinatorial properties of algebraic matroids as derived in [4], the dual of which is non-algebraic. Therefore, it seems to be a good candidate for a negative answer to the old problem whether algebraic matroids are closed under duality (see eg. [8] 6.7.15)

A Pseudoconfiguration of Points without Adjoint

We give an example of a simple oriented matroid D that admits an oriented adjoint. Already any adjoint of the underlying matroid D, however, does itself not admit an adjoint. D arises from the wellknown Non-Desargues-Matroid by a coextension by a coparallel element and, hence, has rank 4. The orientability of D and some of its adjoints follows from an apparantly new oriented matroid construction given in the paper that is a very special case of an amalgam of two copies of one oriented matroid