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## “School of Mathematics”

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Paper   IPM / M / 113
School of Mathematics
Title:   A characterization of uniquely vertex colorable graphs using minimal defining sets
Author(s):
 1 . H. Hajiabolhassan 2 . M.L. Mehrabadi 3 . R. Tusserkani 4 . M. Zaker
Status:   Published
Journal: Discrete Math.
Vol.:  199
Year:  1999
Pages:   233-236
Supported by:  IPM
Abstract:
A defining set (of vertex coloring) of a graph G is a set of vertices S with an assignment of colors to its elements which has a unique completion to a proper coloring of G. We define a minimal defining set to be a defining set which does not properly contain another defining set. If G is a uniquely vertex colorable graph, clearly its minimum defining sets are of size χ(G)−1. It is shown that for a coloring of G, if all minimal defining sets of G are of size χ(G)−1, then G is a uniquely vertex colorable graph.