Yumiko Ohno | Search arXiv

arXiv2021

Difference of facial achromatic numbers between two triangular embeddings of a graph

A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a vertex $k$-coloring such that every triple of $k$-colors appears on the boundary of some face of $G$. The facial $3$-achromatic number $ψ_3(G)$ of $G$ is the maximum integer $k$ such that $G$ has a facial $3$-complete $k$-coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge. For two triangulations $G$ and $G'$ on a surface, $ψ_3(G)$ may not be equal to $ψ_3(G')$ even if $G$ is isomorphic to $G'$ as graphs. Hence, it would be interesting to see how large the difference between $ψ_3(G)$ and $ψ_3(G')$ can be. We shall show that the upper bound for such difference in terms of the genus of the surface.

math.CO↗

arXiv2021

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

In 1967 Harary, Hedetniemi, and Prins showed that every graph $G$ admits a complete $t$-coloring for every $t$ with $χ(G) \le t \le ψ(G)$, 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 Rz\c ażewski (2020) showed that this result fails for hypergraphs by proving that for every integer $k$ with $k\ge 9$, 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 $k\ge 3$ and settle their problems by giving several kinds of $3$-uniform hypergraphs. In particular, we disprove a recent conjecture due to Matsumoto and the third author (2020) who suggested a special family of $3$-uniform hypergraph to satisfy the desired interpolation property.

math.CO↗