B-coloring of Cartesian Product of Trees

Abstract

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. The b-spectrum Sb(G) of a graph G is the set of positive integers k, χ(G) ≤ k ≤ b(G), for which G has a b-coloring using k colors. A graph G is b-continuous if Sb(G) = {χ(G), . . . , b(G)}. It is known that for any two graphs G and H , b(G H) ≥ max{b(G), b(H)}, where stands for the Cartesian product. In this paper, we determine some families of graphs G and H for which b(G H) ≥ b(G) + b(H) − 1. Further if Ti, i = 1, . . . , n are trees with b(Ti) ≥ 3, then b(T1 · · · Tn) ≥ n ∑ i=1 b(Ti)− (n− 1) and Sb(T1 · · · Tn) ⊇ {2, . . . , n ∑ i=1 b(Ti)− (n− 1)}. Also if b(Ti) = Δ(Ti) + 1 for each i, then b(T1 · · · Tn) = Δ(T1 · · · Tn) + 1, and T1 · · · Tn is b-continuous.

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Cite this paper

@inproceedings{Balakrishnan2015BcoloringOC, title={B-coloring of Cartesian Product of Trees}, author={R. Balakrishnan and S. Francis Raj and T. Kavaskar}, year={2015} }