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 $\psi_3(G)$ 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 $G\u27$ on a surface, $\psi_3(G)$ may not be equal to $\psi_3(G\u27)$ even if $G$ is isomorphic to $G\u27$ as graphs. Hence, it would be interesting to see how large the difference between $\psi_3(G)$ and $\psi_3(G\u27)$ can be. We shall show that an upper bound for such difference in terms of the genus of the surface

Nonrepetitive Colourings of Planar Graphs with O(log⁡n)O(\log n) Colours

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$ upper bound

Planar graphs have bounded nonrepetitive chromatic number

A colouring of a graph isnonrepetitiveif for every path of even order, thesequence of colours on the first half of the path is different from the sequence of colours onthe second half. We show that planar graphs have nonrepetitive colourings with a boundednumber of colours, thus proving a conjecture of Alon, Grytczuk, Hałuszczak and Riordan(2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding afixed minor, and graphs excluding a fixed topological minor

Coloring face hypergraphs on surfaces

AbstractThe face hypergraph of a graph G embedded on a surface has the same vertex set as G and its edges are the sets of vertices forming faces of G. A hypergraph is k-choosable if for each assignment of lists of colors of sizes k to its vertices, there is a coloring of the vertices from these lists avoiding a monochromatic edge.We prove that the face hypergraph of the triangulation of a surface of Euler genus g is O(g3)-choosable. This bound matches a previously known lower bound of order Ω (g3). If each face of the graph is incident with at least r distinct vertices, then the face hypergraph is also O(gr)-choosable. Note that colorings of face hypergraphs for r=2 correspond to usual vertex colorings and the upper bound O(g) thus follows from Heawood’s formula. Separate results for small genera are presented: the bound 3 for triangulations of the surface of Euler genus g=3 and the bound 7+36g+496 for surfaces of Euler genus g≥3. Our results dominate the previously known bounds for all genera except for g=4,7,8,9,14