Efficient total face coloring of planar and toroidal maps

Abstract

Let $2\le k\in\mathbb{Z}$. A total coloring of a simple connected regular graph via the color set $ \{0,1,\ldots, k\}$ is said to be {\it efficient} if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. Focus is set upon planar and toroidal maps. Each such map is said to cover its induced graph. An efficient total coloring of one such graph induces an efficient total face coloring of its covering combinatorial map if it assigns a vertex-and-edge $k$-color set to the boundary cycle of each of its faces, with the consequently missing color in $\{0,1,\ldots,k\}$ assigned to the face itself, so that the two adjacent faces along any edge are assigned different colors.