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Smallest defining number of r-regular k-chromatic graphs: R ≠ k

Mahmoodian, E. S ; Sharif University of Technology | 2006

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  1. Type of Document: Article
  2. Publisher: 2006
  3. Abstract:
  4. In a given graph G, a set S of vertices with an assignment of colors is a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a χ(G)-coloring of the vertices of G. A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by d(G, χ)- Let d(n, r, χ = k) be the smallest defining number of all r-regular k-chromatic graphs with n vertices. Mahmoodian and Mendelsohn (1999) proved that for each n and each r ≥ 4, d(n, r, χ = 3) = 2. They raised the following question: Is it true that for every k, there exist n0(k) and r0(k), such that for all n ≥ n0(k) and r ≥ r0(k) we have d(n, r, χ =k) = k -1? We show that the answer to this question is positive, and we prove that for a given k and for all n ≥ 3k, if r ≥ 2(k-1) then d(n, r, χ = k) = k-1
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  7. Source: Ars Combinatoria ; Volume 78 , 2006 , Pages 211-223 ; 03817032 (ISSN)
  8. URL: https://dblp.org/db/journals/arscom/arscom78.html#MahmoodianOS06