The Complexity of Vertex Coloring Problems in Uniform Hypergraphs with High Degree

Abstract

In this note we consider the problem of deciding whether a given r-uniform hypergraph H with minimum vertex degree at least c(|V(H)|-1 r-1), has a vertex 2-coloring and a strong vertex k-coloring. Motivated by an old result of Edwards for graphs, we summarize what can be deduced from his method about the complexity of these problems for hypergraphs. We obtain the first optimal dichotomy results for 2-colorings of 3- and 4-uniform hypergraphs according to the value of c. In addition, we determine the computational complexity of strong k-colorings of 3-uniform hypergraphs for some c, leaving a gap which vanishes as k→∞.