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

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