b-Chromatic Sum and b-Continuity Property of Some Graphs

A b-coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] 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 [Formula: see text], denoted by [Formula: see text], is the largest integer [Formula: see text] such that [Formula: see text] has a b-coloring with [Formula: see text] colors. The b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text], is introduced and it is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for any [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. A graph [Formula: see text] is b-continuous, if it admits a b-coloring with [Formula: see text] colors, for every [Formula: see text]. In this paper, the [Formula: see text]-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.

On the b-Chromatic Sum of Mycielskian of Km, n, Kn and Cn

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,…, vn} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u1, u2, …, un} joining u to each vertex ui(1 ≤ i ≤ n) and joining ui to each neighbour of vi 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.

Task Scheduling Problem Using Fuzzy Graph

The concept of obtaining fuzzy sum of fuzzy colorings problem has a natural application in scheduling theory. The problem of scheduling N jobs on a single machine and obtain the minimum value of the job completion times is equivalent to finding the fuzzy chromatic sum of the fuzzy graph modeled for this problem. The aim of this paper is to solve task scheduling problems using fuzzy graph.

Certain chromatic sums of some cycle-related graph classes

Let [Formula: see text] be a certain type of proper [Formula: see text]-coloring of a given graph [Formula: see text] and [Formula: see text] denote the number of times a particular color [Formula: see text] is assigned to the vertices of [Formula: see text]. Then, the coloring sum of a given graph [Formula: see text] with respect to the coloring [Formula: see text], denoted by [Formula: see text] is defined to be [Formula: see text]. The coloring sums such as [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, [Formula: see text]-chromatic sum, etc. are some of these types of coloring sums that have been studied recently. Motivated by these studies on certain chromatic sums of graphs, in this paper, we study certain chromatic sums for some standard cycle-related graphs.

b-Chromatic sum of a graph

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 [Formula: see text], is the maximum integer [Formula: see text] such that G admits a b-coloring with [Formula: see text] colors. In this paper we introduce a new concept, the b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text] and is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for all [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. Also obtained the b-chromatic sum of paths, cycles, wheel graph, complete graph, star graph, double star graph, complete bipartite graph, corona of paths and corona of cycles.