Neighborhood-restricted [≤2][≤2]-achromatic colorings

A (closed) neighborhood-restricted [≤2][≤2]-coloring of a graph GG is an assignment of colors to the vertices of GG such that no more than two colors are assigned in any closed neighborhood, that is, for every vertex vv in GG, the vertex vv and its neighbors are in at most two different color classes. The [≤2][≤2]-achromatic number   is defined as the maximum number of colors in any [≤2][≤2]-coloring of GG. We study the [≤2][≤2]-achromatic number. In particular, we improve a known upper bound and characterize the extremal graphs for some other known bounds.