Brooks type results for conflict-free colorings and {a, b}-factors in graphs

A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each r-uniform hypergraph of maximum vertex degree ∆ has a conflict-free coloring with at most k colors. As shown by Tardos and Pach, similarly to a classical Brooks’ type theorem for hypergraphs, f(r,∆) ≤ ∆+1. Compared to Brooks’ theorem, according to which there is only a couple of graphs/hypergraphs that attain the ∆+1 bound, we show that there are several infinite classes of uniform hypergraphs for which the upper bound is attained. We give better upper bounds in terms of ∆ for large ∆ and establish the connection between conflict-free colorings and so-called {t, r−t}-factors in r-regular graphs. Here, a {t, r − t}-factor is a factor in which each degree is either t or r − t. Among others, we disprove a conjecture of Akbari and Kano [1] stating that there is a {t, r − t}-factor in every r-regular graph for odd r and any odd t < r 3 .