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Extending graph total colorings to cell complexes

Let $2\le k\in\mathbb{Z}$. A total coloring of a simple connected regular graph via 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. In this work, focus is set upon 2-cell complexes whose 1-skeletons, namely their induced 1-cell complexes, are toroidal graphs. Each such 2-cell complex is said to cover its induced 1-skeleton. An efficient total coloring of one such skeleton induces an efficient total cell coloring of its covering 2-cell complex if it assigns a vertex-and-edge $k$-color set to the border skeleton of each of its 2-cells, with the consequently missing color in $\{0,1,\ldots,k\}$ assigned to the 2-cell itself, so that the two adjacent 2-cells along any 1-cell are assigned different colors. Applications are given for plane tilings, cycle products, toroidal triangulations, honeycombs and star-of-David tilings.