Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 54: Issue 1
On two conjectures on b-coloring of graph products
Authors: S. Francis Raj 1 and T. Kavaskar 2
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  • 1 Pondicherry University, Pondicherry–605014, India
  • 2 CIT campus, Taramani, Chennai-600113, India
DOI:
https://doi.org/10.1556/012.2017.54.1.1358
Page Count:
141–149
Publication Date:
01 Mar 2017
Online Publication Date:
Mar 2017
Article Category:
Research Article
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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. It was conjectured in [10], that for any two graphs G and H, b(G[H]) ≦ b(G) − 1|V (H)| + Δ(H) + 1 and b(G ⊠ H) ≦ max {b(G)(Δ(H) + 1), b(H) Δ(G) + 1)}, where G[H] and G ⊠ H denotes the lexicographic product and the strong product of G and H, respectively. In this paper, we disprove both conjectures.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896
Keywords:
Primary 05C15; b-chromatic number; strong product; lexicographic product

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