Some k-fold edge-graceful labelings of (p; p ¡ 1)-graphs
Wai Chee Shiu, S. M. Lee, Karl Schaffer · 2001
An edge-graceful (p, q)-graph G = (V, E) is a graph with p vertices and q edges for which there is a bijection f: E → {1, 2,..., q} such that the induced mapping f +: V → Zp defined by f + (u) ≡ � f(uv) (mod p), for u ∈ V, is a bijection. In this paper, some results on edge-uv∈E gracefulness of trees are extended to k-fold graphs based on graphs with p vertices and p − 1 edges. A k-fold multigraph G[k] derived from a graph G is one in which each edge of G has been replaced by k parallel edges with the same vertices as the original edge. Certain classes of k-fold multigraphs derived from paths, combs, and spiders are shown to be edge-graceful, as well as other graphs constructed by combining these graphs in specified ways. 1