The existence of uniform hypergraphs for which the interpolation property of complete coloring fails

Abstract

In 1967 Harary, Hedetniemi, and Prins showed that every graph G admits a complete t-coloring for every t with χ(G)≤t≤ψ(G), where χ(G) denotes the chromatic number of G and ψ(G) denotes the achromatic number of G which is the maximum number r for which G admits a complete r-coloring. Recently, Edwards and Rza̧żewski (2020) showed that this result fails for hypergraphs by proving that for every integer k with k≥9, there exists a k-uniform hypergraph H with a complete χ(H)-coloring and a complete ψ(H)-coloring, but no complete t-coloring for some t with χ(H)<t<ψ(H). They also asked whether there would exist such an example for 3-uniform hypergraphs and posed another problem to strengthen their result. In this paper, we generalize their result to all cases k with k≥3 and settle their problems by giving several examples of 3-uniform hypergraphs. In particular, we disprove a recent conjecture due to Matsumoto and Ohno (2020) who suggested a special family of 3-uniform hypergraph to satisfy the desired interpolation property.