(untitled) (European Journal of Pure and Applied Mathematics, 2025)
A b-coloring of a graph G is a proper coloring such that there exists a vertex in each color class that is adjacent to at least one vertex in other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has b-coloring with k colors. The b-chromatic sum of graph G, denoted by φ′(G), is defined as the minimum sum of the colors c(v) of v for all v ∈ V where c is a b-coloring with colors 1, 2, . . . , φ(G). In the work, we improve the bounds on the b-chromatic sum given by Lisna and Sunitha [4]. We give the b-chromatic sum of the Mycielskian of a path μ(Pn) when n = 7, 9 and n ≥ 16. For the case 10 ≤ n ≤ 15, we give bounds on φ′(μ(Pn)).