Colorings and nowhere-zero flows of graphs in terms of Berlekamp's Switching Game

We work with a unifying linear algebra formulation for nowhere-zero flows and colorings of graphs and matrices. Given a subspace (code) U ≤ ℤ_kⁿ - e.g. the bond or the cycle space over ℤ_k of an oriented graph - we call a nowhere-zero tuple f ε ℤ_kⁿ a flow of U if f is orthogonal to U. In order to detect flows, we view the subspace U as a light pattern on the n-dimensional Berlekamp Board ℤ_kⁿ with kⁿ light bulbs. The lights corresponding to elements of U are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. The core result of this paper is that the subspace U has a flow if and only if the light pattern U cannot be switched off. In particular, a graph G has a nowhere-zero k-flow if and only if the ℤ_k-bond space of G cannot be switched off. It has a vertex coloring with k colors if and only if a certain corresponding code over ℤ_k cannot be switched off. Similar statements hold for Tait colorings, and for nowhere-zero points of matrices. Studying different normal forms to equivalence classes of light patterns, we find various new equivalents, e.g., for the Four Color Problem, Tutte's Flow Conjectures and Jaeger's Conjecture. Two of our equivalents for colorability and existence of nowhere zero flows of graphs include as special cases results by Matiyasevich, by Baĺazs Szegedy, and by Onn. Alon and Tarsi's sufficient condition for k-colorability also arrives, remarkably, as a generalized full equivalent.