The line completion number of hypercubes

Abstract

In 1992, Bagga, Beineke, and Varma introduced the concept of the super line graph of index r of a graph G, denoted by L r (G). The vertices of L r (G) are the r -subsets of E(G), and two vertices S and T are adjacent if there exist s ∈ S and t ∈ T such that s and t are adjacent edges in G. They also defined the line completion number l c (G) of graph G to be the minimum index r for which L r (G) is complete. They found the line completion number for certain classes of graphs. In this paper, we find the line completion number of hypercube Q n for every n. c

On Several Parameters of Super Line Graph L2(G)

“…In 2008, Li, Li, and Zhang [15] showed that if G has no isolated edges, then L 2 (G) is path-comprehensive, and that if G has at most one isolated edge, then L 2 (G) is vertex-pancyclic, answering a question posed by Bagga, Beineke, and Varma [14]. We refer to [16][17][18][19][20][21][22][23][24][25][26] for more results on super line graphs.…”

“…The line completion number lc(G) of a graph G is the least index r for which L r (G) is complete. This notion was investigated in [22][23][24][25][26]. For a graph G without an isolated vertex,…”

Order Structure of Good Sets in Hypercube

“…Tapadia and Waphare proved graph theoretic properties like good sets are partial cubes, Hamiltonian, bipancyclic, etc. [7]. One can refer for more properties of good sets in [8][9][10][11][12].…”