On the Dominator Colorings in Bipartite Graphs

“…Hence for chordal graphs max{χ(G), γ(G) + 1} ≤ χ d (G) ≤ χ(G) + γ(G), and both the upper and lower bounds are sharp. For a bipartite graph G, the dominator chromatic number takes one of the three values γ(G), 1550043 On the dominator coloring in proper interval graphs and block graphs γ(G) + 1 or γ(G) + 2 [7]. A necessary and sufficient condition for bipartite graphs of order at least 4 and χ d (G) = γ(G) is also given in [7].…”

Section: Introductionmentioning

confidence: 99%

“…For a bipartite graph G, the dominator chromatic number takes one of the three values γ(G), 1550043 On the dominator coloring in proper interval graphs and block graphs γ(G) + 1 or γ(G) + 2 [7]. A necessary and sufficient condition for bipartite graphs of order at least 4 and χ d (G) = γ(G) is also given in [7]. For a tree T , the dominator chromatic number takes one of the two values γ(T ) + 1 or γ(T ) + 2 [7].…”

Section: Introductionmentioning

confidence: 99%

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On the dominator coloring in proper interval graphs and block graphs

Panda

Pandey

2015

Discrete Math. Algorithm. Appl.

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In a graph G = (V, E), a vertex v dominates a vertex w if either v = w or v is adjacent to w. A subset of vertex set V that dominates all the vertices of G is called a dominating set of graph G. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by γ(G). A proper coloring of a graph G is an assignment of colors to the vertices of G such that any two adjacent vertices get different colors. The minimum number of colors required for a proper coloring of G is called the chromatic number of G and is denoted by χ(G). A dominator coloring of a graph G is a proper coloring of the vertices of G such that every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of G is called the dominator chromatic number of G and is denoted by χ d (G). In this paper, we study the dominator chromatic number for the proper interval graphs and block graphs. We show that every proper interval graph G satisfies, and these bounds are sharp. For a block graph G, where one of the end block is of maximum size, we show thatWe also characterize the block graphs with an end block of maximum size and attaining the lower bound.

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“…Dominating Set and Coloring have a number of applications and this has led to the algorithmic study of numerous variants of these problems. A number of basic combinatorial and algorithmic results on DC have been obtained in [2], [3], [4] and [5]. A systematic study of dominator coloring problem from the perspective of algorithms and complexity was initiated in [6].…”

Section: Introductionmentioning

confidence: 99%