This paper deals with vertex-magic total labellings of graphs. Earlier work by many authors has shown many infinite families of graphs to admit such labelings. The fact that many of these graphs are regular led MacDougall to conjecture that all non-trivial regular graphs are vertexmagic. Previously Gray and MacDougall showed that all odd-order rregular graphs (r ≥ 2) of order up to v = 19 are v...

Let G = (V, E) be a (p, q)-graph. Let f : E → {1, 2, . . . , q} be a bijection. The induced mapping f : V → Zp of f is defined by f (u) ≡ ∑ uv∈E f(uv) (mod p) for u ∈ V . If f is a bijection, then G is called edge-graceful. In this paper, we investigate the edge-gracefulness of the composition of paths with null graphs Pm ◦ Nn, where there are mn vertices and (m − 1)n edges. We show that P3 ◦ N...

For a simple path Pr on r vertices, the square of Pr is the graph P 2 r on the same set of vertices of Pr, and where every pair of vertices of distance 2 or less in Pr are connected by an edge. Given a (p, q)-graph G with p vertices and q edges, and an integer k ≥ 0, G is said to be k-edge-graceful if the edges can be labeled by k, k+1, . . . , k+q−1 so that the induced vertex sums (mod p) are ...

A (p, q) connected graph is edge-odd graceful graph if there exists an injective map f: E(G) → {1, 3, ..., 2q-1} so that induced map f+: V(G) → {0, 1,2, 3, ..., (2k-1)}defined by f+(x)  f(x, y) (mod 2k), where the vertex x is incident with other vertex y and k = max {p, q} makes all the edges distinct.

R. B. Gnanajothi, Topics in graph theory, Ph. D. thesis, Madurai Kamaraj University, India, 1991. E. M. Badr, On the Odd Gracefulness of Cyclic Snakes With Pendant Edges, International journal on applications of graph theory in wireless ad hoc networks and sensor networks (GRAPH-HOC) Vol. 4, No. 4, December 2012. E. M. Badr, M. I. Moussa & K. Kathiresan (2011): Crown graphs and subdivision of l...