Realizability of Polytopes as a Low Rank Matrix Completion Problem

This article gives necessary and sufficient conditions for a relation to be the containment relation between the facets and vertices of a polytope. Also given here, are a set of matrices parameterizing the linear moduli space and another set parameterizing the projective moduli space of a combinatorial polytope

GAME CHROMATIC NUMBER OF CARTESIAN PRODUCT GRAPHS

The game chromatic number χg is considered for the Cartesian product G ✷ H of two graphs G and H. We determine exact values of χg(G✷H) when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number χg(G✷H) is not bounded from above by a function of game chromatic numbers of graphs G and H. An analogous result is proved for the game coloring number colg(G✷H) of the Cartesian product of graphs

NONREPETITIVE COLORINGS OF TREES

A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by π(G). A famous theorem of Thue asserts that π(P) = 3 for any path P with at least 4 vertices. In this paper we study the Thue chromatic number of trees. In view of the fact that π(T) is bounded by 4 in this class we aim to describe the 4-chromatic trees. In particular, we study the 4-critical trees which are minimal with respect to this property. Though there are many trees T with π(T) = 4 we show that any of them has a sufficiently large subdivision H such that π(H) = 3. The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees. By a similar argument we prove that any tree has a subdivision which can be edge-colored by at most ∆ + 1 colors without repetitions on paths