Some New Characterizations of Graph Colorability and of Blocking Sets of Projective Spaces

Uwe Schauz

doi:10.37236/3767

Abstract

Let $G=(V,E)$ be a graph and $q$ be an odd prime power. We prove that $G$ possess a proper vertex coloring with $q$ colors if and only if there exists an odd vertex labeling $x\in F_q^V$ of $G$. Here, $x$ is called odd if there is an odd number of partitions $\pi=\{V_1,V_2,\dotsc,V_t\}$ of $V$ whose blocks $V_i$ are $G$-bipartite and $x$-balanced, i.e., such that $G|_{V_i}$ is connected and bipartite, and $\sum_{v\in V_i}x_v=0$. Other new characterizations concern edge colorability of graphs and, on a more general level, blocking sets of projective spaces. Some of these characterizations are formulated in terms of a new switching game.

Highlights

We prove that G possess a proper vertex coloring with q colors if and only if there exists an odd vertex labeling x ∈ FqV of G
Other new characterizations concern edge colorability of graphs and, on a more general level, blocking sets of projective spaces. Some of these characterizations are formulated in terms of a new switching game
The results in this paper are based on the study of a new switching game (Section 2)

Summary

Introduction

The results in this paper are based on the study of a new switching game (Section 2). In some of the applications of such tools, it can be important to know that the number of colorings of a graph is nonzero modulo r. One can sometimes use the Combinatorial Nullstellensatz to prove the colorability of some graphs or to generalize colorability results to list coloring results, as e.g. in [Sch, Sec. 5] In such approaches, one usually has to show that a certain coefficient in a certain polynomial is nonzero. It is conceivable that one can find ways to employ this, for example in connection with the Combinatorial Nullstellensatz

Switching Games

Blocking Sets

Graph Colorings