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, subj...

There are many long-standing conjectures related with some labellings of trees. It is important to connect labellings that are related with conjectures. We find some connections between known labellings of simple graphs.

A rooted tree with diameter D is said to have an even degree sequence if every vertex has even degree except for one root and the leaves, which are in the last level D/2 . The degree sequence is said to be quasi even if every vertex has even degree except for one root, every vertex in level D/2 − 1 and the leaves, which are in the last level D/2 . Hrnčiar and Haviar [P. Hrnčiar, A. Haviar, All ...

A difference vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy the weight |f(x)− f(y)| . A difference vertex labeling f of a graph G of size n is odd-graceful if f is an injection from V (G) to {0, 1, ..., 2n − 1} such that the induced weights are {1, 3, ..., 2n − 1}. We show here that any forest whose components are caterpillars is odd...

Here we denote a diameter six tree by (a0; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr), where a0 is the center of the tree; ai, i = 1, 2, . . . ,m, bj , j = 1, 2, . . . , n, and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to a0; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful...

A graceful labelling of a graph with n edges is a vertex labelling where the induced set of edge weights is {1, . . . , n}. A near graceful labelling is almost the same, the difference being that the edge weights are {1, 2, . . . , n − 1, n + 1}. In both cases, the weight of an edge is the absolute difference between its two vertex labels. Rosa [8] in 1988 conjectured that all triangular cacti ...

Here we denote a diameter six tree by (c; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr), where c is the center of the tree; ai, i = 1, 2, . . . ,m, bj, j = 1, 2, . . . , n, and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to c; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful lab...

We exhibit a graceful labelling for each generalised Petersen graph P8t,3 with t ≥ 1. As an easy consequence, we obtain that for any fixed t the corresponding graph is the unique starter graph for a cyclic edgedecomposition of the complete graph K2t+1. Due to its extreme versatility, the technique employed looks promising for finding new graceful labellings, not necessarily involving generalise...

An a-labeling of a bipartite graph G with n edges easily yields both a cyclic G-decomposition of Kn,n and of K2nx+1 for all positive integers x. A ,B-Iabeling (or graceful labeling) of G yields a cyclic decomposition of K2n+1 only. It is well-known that certain classes of trees do not have a-Iabelings. In this article, we introduce the concept of a near a-labeling of a bipartite graph, and prov...