An Alon–Tarsi Style Theorem for Additive Colorings

Abstract

We first give a proof of the Alon–Tarsi list coloring theorem that differs from Alon and Tarsi’s original. We use the ideas from this proof to obtain the following result, which is an additive coloring analog of the Alon–Tarsi Theorem: Let G be a graph and let D be an orientation of G. We introduce a new digraph \(\mathcal {W}(D)\), such that if the out-degree in D of each vertex v is \(d_v\), and if the number of Eulerian subdigraphs of \(\mathcal {W}(D)\) with an even number of edges differs from the number of Eulerian subdigraphs of \(\mathcal {W}(D)\) with an odd number of edges, then for any assignment of lists L(v) of \(d_v+1\) positive integers to the vertices of G, there is an additive coloring of G assigning to each vertex v an element from L(v). As an application, we prove an additive list coloring result for tripartite graphs G such that one of the color classes of G contains only vertices whose neighborhoods are complete.