Regular number of Middle graph of a graph

Abstract

For any (p ,q) graph G, the middle graph of a graph G, is denoted by M(G), is a graph whose vertex set is V(G)?E(G), and two vertices are adjacent if they are adjacent edges of G or one is a vertex and other is an edge incident with it. The regular number of the M(G) is the minimum number of subsets into which the edge set of M(G) should be partitioned so that the subgraph induced by each subset is regular and is denoted by r_M (G). In this paper some results on regular number of r_M (G) were obtained and expressed in terms of elements of G.