Spectrum of Mixed Bi-uniform Hypergraphs

A mixed hypergraph is a triple H = (V, C, D), where V is a set of vertices, C and D are sets of hyperedges. A vertex-coloring of H is proper if C-edges are not totally multicolored and D-edges are not monochromatic. The feasible set S(H) of H is the set of all integers, s, such that H has a proper coloring with s colors. Bujt´as and Tuza [Graphs and Combinatorics 24 (2008), 1–12] gave a characterization of feasible sets for mixed hypergraphs with all C- and D-edges of the same size r, r ≥ 3. In this note, we give a short proof of a complete characterization of all possible feasible sets for mixed hypergraphs with all C-edges of size and all D-edges of size m, where , m ≥ 2. Moreover, we show that for every sequence (r(s))n s= , n ≥ , of natural numbers there exists such a hypergraph with exactly r(s) proper colorings using s colors, s = , . . . , n, and no proper coloring with more than n colors. Choosing = m = r this answers a question of Bujt´as and Tuza, and generalizes th...

• Type: Journal
• Year: 2016
• Where: GC
• Authors: Maria Axenovich, Enrica Cherubini, Torsten Ueckerdt
• Keywords: Applied Computing, GC 2016