Discussiones Mathematicae Graph Theory 32(1) (2012) 81-90
doi: 10.7151/dmgt.1587

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Recognizable Colorings of Cycles and Trees

Michael J. Dorfling

Department of Mathematics
University of Johannesburg
Johannesburg, South Africa

Samantha Dorfling

Department of Mathematics and Applied Mathematics
University of the Free State
Bloemfontein, South Africa

Abstract

For a graph G and a vertex-coloring c:V(G)→{1,2, …,k}, the color code of a vertex v is the (k+1)-tuple (a0,a1, …,ak), where a0 = c(v), and for 1 ≤ i ≤ k, ai is the number of neighbors of v colored i. A recognizable coloring is a coloring such that distinct vertices have distinct color codes. The recognition number of a graph is the minimum k for which G has a recognizable k-coloring. In this paper we prove three conjectures of Chartrand et al. in [8] regarding the recognition number of cycles and trees.

Keywords: recognizable coloring, recognition number

2010 Mathematics Subject Classification: 05C15, 05C70.

References

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Received 16 July 2010
Revised 25 January 2011
Accepted 25 January 2011