On Modular Edge-Graceful Graphs

Abstract

Let G be a connected graph of order ${n\ge 3}$ and size m and ${f:E(G)\to \mathbb{Z}_n}$ an edge labeling of G. Define a vertex labeling ${f': V(G)\to \mathbb{Z}_n}$ by ${f'(v)= \sum_{u\in N(v)}f(uv)}$ where the sum is computed in ${\mathbb{Z}_n}$ . If f′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A graph G is modular edge-graceful if G contains a modular edge-graceful spanning tree. Several classes of modular edge-graceful trees are determined. For a tree T of order n where ${n\not\equiv 2 \pmod 4}$ , it is shown that if T contains at most two even vertices or the set of even vertices of T induces a path, then T is modular edge-graceful. It is also shown that every tree of order n where ${n\not\equiv 2\pmod 4}$ having diameter at most 5 is modular edge-graceful.