# On a Sufficient and Necessary Condition for Graph Coloring

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Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: for given m and n with m < n, m is adjacent to n if n has a 1 in the mth position of its binary expansion. It is well known that R is a universal graph in the set...

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Let G be a vertex colored graph. The minimum number Ï‡(G) of colors needed for coloring of a graph G is called the chromatic number. Recently, Adiga et al. [1] have introduced the concept of color energy of a graph Ec(G) and computed the color energy of few families of graphs with Ï‡(G)...

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A proper k-coloring is called a b-coloring if there exits a vertex (b-vertex) that has neighbour(s) in all other k - 1 color classes. The largest integer k for which G admits a b-coloring is called the b-chromatic number denoted as Ï†(G). If b-coloring exists for every integer k satisfying...

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Rainbow connection number, rc( G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (note that the coloring need not be proper). In this paper we study the...

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The general vertex-distinguishing total chromatic number of a graph G is the minimum integer K, for which the vertices and edges of G are colored using K colors such that any two vertices have distinct sets of colors of them and their incident edges. In this paper, we figure out the exact value...

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No abstract available.

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No abstract available.