Achromatic number and facial achromatic number of connected locally-connected graphs

A graph is locally-connected if the neighborhood of each vertex induces a connected graph. It is well known that a triangulation on a closed surface is locally-connected, and some results for triangulations were generalized to those for connected locally-connected graphs. In this paper, we extend two characterization theorems of triangulations for a complete coloring and a facial complete coloring, which are vertex colorings with constraints on the appearance of color tuples, to those of connected locally-connected graphs. Moreover, we also investigate the relation between the corresponding invariants and the number of independent elements.