Formula omitted]-labeling of supersubdivided connected graph plus an edge

Abstract

Rosa, in his classical paper (Rosa, 1967) introduced a hierarchical series of labelings called ρ,σ,β and α labeling as a tool to settle Ringel’s Conjecture which states that if T is any tree with q edges then the complete graph K2q+1 can be decomposed into 2q+1 copies of T. Inspired by the result of Rosa, many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco, El-Zanati and Vanden Eynden introduced γ-labeling as a stronger version of ρ-labeling. A function h defined on the vertex set of a graph G with q edges is called a γ-labeling if(i) h is a ρ-labeling of G,(ii) G is tripartite with vertex tripartition (A,B,C) with C={c} and b̄∈B such that (b̄,c) is the unique edge joining an element of B to c,(iii) for every edge (a,v)∈E(G) with a∈A, h(a)<h(v),(iv) h(c)−h(b̄)=q.Further, Blinco et al. proved a significant result that if a graph G with q edges admits a γ-labeling, then the complete graph K2cq+1 can be cyclically decomposed into 2cq+1 copies of the graph G, where c is any positive integer. Motivated by the result of Blinco et al., we show that a new family of almost bipartite graphs each admits γ-labeling. The new family of almost bipartite graphs is defined based on the supersubdivision graph of certain connected graph. Supersubdivision graph of a graph was introduced by Sethuraman and Selvaraju in Sethuraman and Selvaraju (2001). A graph is said to be a supersubdivision graph of a graph G with q edges, denoted SSD(G) if SSD(G) is obtained from G by replacing every edge ei of G by a complete bipartite graph K2,mi, 1≤i≤q, (where mi may vary for each edge ei) in such a way that the ends of ei are identified with the 2 vertices of the vertex part having two vertices of the complete bipartite graph of K2,mi after removing the edge ei of G. In the graph SSD(G), the vertices which originally belong to the graph G are called the base vertices of SSD(G) and all the other vertices of SSD(G) are called the non-base vertices of SSD(G). More precisely, we prove that if G is a connected graph containing a cycle Ck, where k≥6 and having a vertex of degree two with one of its adjacent vertices of degree one and its other adjacent vertex is of degree at least two, then certain supersubdivision graph of the graph G, SSD(G) plus an edge eˆ admits γ-labeling, where eˆ is added between a suitably chosen pair of non-base vertices of the graph SSD(G). Also, we discuss a related open problem.