Variation of Graceful Labeling on Disjoint Union of two Subdivided Shell Graphs

A shell graph is the join of a path P k of 'k' vertices and K 1. A subdivided shell graph can be constructed by subdividing the edges in the path of the shell graph. In this paper we prove that the disjoint union of two subdivided shell graphs is odd graceful and also one modulo three graceful. 1. Introduction A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. In 1967 Rosa[10] introduced the labeling method called β-valuation as a tool for decomposing the complete graph into isomorphic sub-graphs. Later on, this β-valuation was renamed as graceful labeling by Golomb [9]. A graceful labeling of a graph G with 'q' edges and vertex set V is an injection f :V(G) → {0,1,2,….q} with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label |f(u) – f(v)|. A graph which admits a graceful labeling is called a graceful graph. A variation of graceful labeling is odd-graceful labeling. This was introduced by Gnanajothi [8] in the year 1991. She defined a graph G with q edges to be odd-graceful if there is an injection f :V(G) → {0, 1, 2,. .. , 2q−1} such that, when each edge xy is assigned the label |f(x)−f(y)|, the resulting edge labels are {1, 3, 5,. .. , 2q−1}. She proved many graphs as odd-graceful: paths P n , C n if and only if n is even, K m,n , combs P n Θ K 1 , books, crowns C n Θ K 1 if and only if n is even, , the one-point union of copies of C 4, C n × K 2 if and only if n is even, caterpillars, rooted trees of height 2. Eldergill [5] generalized Gnanajothi's result on stars. Barrientos [2] has proved the following graphs are odd-graceful: every forest whose components are caterpillars, every tree with diameter at most five and all disjoint unions of caterpillars. Seoud, Diab, and Elsakhawi [12] have shown that a connected complete r-partite graph is odd-graceful if and only if r = 2.