On some covering graphs of a graph

Abstract

For a graph G with vertex set V (G) = [v1; v2; ...; vn], let S be the covering set of G having the maximum degree over all the minimum covering sets of G. Let NS[v] = [u ε S : uv ε E(G)] [v]g be the closed neighbourhood of the vertex v with respect to S: We define a square matrix AS(G) = (aij); by aij = 1; if |NS[vi] ∩ NS[vj]| ≥ 1, i ≠ j and 0, otherwise. The graph GS associated with the matrix AS(G) is called the maximum degree minimum covering graph (MDMC-graph) of the graph G. In this paper, we give conditions for the graph GS to be bipartite and Hamiltonian. Also we obtain a bound for the number of edges of the graph GS in terms of the structure of G. Further we obtain an upper bound for covering number S of GS in terms of the covering number (independence number) of G.