Loading...
On defining numbers of k-chromatic k-regular graphs
Soltankhah, N ; Sharif University of Technology | 2005
178
Viewed
- Type of Document: Article
- Publisher: 2005
- Abstract:
- 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). We study the defining number of regular graphs. Let d(n,r, Χ -k) be the smallest defining number of all r-regular k-chromatic graphs with n vertices, and f(n, k) = k-2/2(k-1)n + 2+(k-2)(k-3)/2(k-1). Mahmoodian and Mendelsohn (1999) determined the value of d(n, k, Χ = k) for all k ≤ 5, except for the case of (n, k) = (10,5). They showed that d(n, k, Χ = k) = [f(n, k)] for k ≤ 5. They raised the following question: Is it true that for every k, there exists n 0(k) such that for all n ≥ n0(k), we have d(n, k, Χ = k) = [f(n, k)]? Here we determine the value of d(n, k, Χ = k) for each k in some congruence classes of n. We show that the answer for the question above, in general, is negative. Also here, for k = 6 and k = 7 the value of d(n, k, Χ = k) is determined except for one single case, and it is shown that d(10, 5, Χ = 5) = 6
- Keywords:
- Colorings ; Uniquely completable pre-coloring ; Regular graphs ; Defining sets
- Source: Ars Combinatoria ; Volume 76 , 2005 , Pages 257-276 ; 03817032 (ISSN)
- URL: https://www.researchgate.net/publication/220620028_On_defining_numbers_of_k-chromatic_k-regular_graphs