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A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by $\phi \left(G\right)$, is the maximal integer k such that G has a b-coloring with k colors. In this paper, the b-chromatic numbers of the coronas of cycles, star graphs and wheel graphs with different numbers of vertices, respectively, are obtained. Also the bounds for the b-chromatic number of corona of any two graphs is discussed.

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v₁, v₂,…, v_n} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u₁, u₂, …, u_n} joining u to each vertex u_i(1 ≤ i ≤ n) and joining u_i to each neighbour of v_i in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

A b-coloring of a graph $G$ is a proper coloring of the vertices of $G$ such that there exist a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph $G$, denoted by $\phi (G)$, is the largest integer $k$ such that $G$ has a b-coloring with $k$ colors. The b-chromatic sum of a graph $G(V,E)$, denoted by ${\phi }^{\prime }(G)$, is introduced and it is defined as the minimum of sum of colors $c(v)$ of $v$ for any $v\in V$ in a b-coloring of $G$ using $\phi (G)$ colors. A graph $G$ is b-continuous, if it admits a b-coloring with $t$ colors, for every $t=\chi (G),\dots ,\phi (G)$. In this paper, the $b$-continuity property of corona of two cycles, corona of two star graphs and corona of two wheel graphs with unequal number of vertices is discussed. The b-continuity property of corona of any two graphs with same number of vertices is also discussed. Also, the b-continuity property of Mycielskian of complete graph, complete bipartite graph and paths are discussed. The b-chromatic sum of power graph of a path is also obtained.