On conjectures on the defining set of (vertex) graph colourings

In a given graph G = (V, E), a set of vertices S 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 chi(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, chi). In Combinatorics, Graph Theory and Algorithms (1999), 461-467, Mandian et al. have studied d(C(m)x K-3, chi) d(C-m x K-4, chi) and d(C(m)x K-5, chi). They have conjectured: (a) d(C(m)x K-5, chi) = 2m + 1 for in odd; and (b) d(C(m)x K-4, chi) = m + 1. In this paper we disprove conjecture (a) for m(>= 5) odd and prove conjecture (b).