On defining numbers of vertex colouring of regular graphs

AbstractIn a given graph G, a set S of vertices with an assignment of colours to them is a defining set of the vertex colouring of G, if there exists a unique extension of the colours of S to a χ(G)-colouring of the vertices of G.A defining set with minimum cardinality is called a minimum defining set (of vertex colouring) and its cardinality, the defining number, is denoted by d(G, χ). Mahmoodian et al., have studied this concept. Here we study the defining numbers of regular graphs. Among other results the exact value of d(n, r, χ = r), the minimum defining number of all r-regular r-chromatic graphs with n vertices is determined, for r = 2, 3, 4, and 5