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

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 ∈ FV q of G. Here x is called odd if there is an odd number of partitions π = {V1, V2, . . . , Vt} of V whose blocks Vi are G-bipartite and x-balanced, i.e., such that G|Vi is connected and bipartite, and ∑ v∈Vi xv = 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.