Achromatic number of some classes of digraphs

Abstract

Let D be a directed graph with n vertices and m arcs. A function g: V (D) →{1, 2, ⋯, k} where k ≤ n is called a complete coloring of D if and only if for every arc uv of D, the ordered pair (g(u), g(v)) appears at least once. If the pair (i, i) is not assigned, then g is called a proper complete coloring of D. The maximum k for which D admits a proper complete coloring is called the achromatic number of D and is denoted by ψ<inf>c</inf> →(D). We obtain the upper bound for the achromatic number of digraphs and regular digraphs and investigate the same for some classes of digraphs such as unipath, unicycle, circulant digraphs, etc. © 2024 World Scientific Publishing Company.