Edge consecutive gracefulness of a graph

Abstract

Let G=(V,E) be a (p,q)-graph without isolated vertices. The graph G is said to be graceful if there exists an injection f:V→{0,1,…,q} such that for the induced edge labeling gf defined on E by gf(uv)=|f(u)−f(v)|, the edge labels are distinct. The gracefulness grac(G) of G is the smallest positive integer k for which there exists an injective function f:V→{0,1,2,…,k} such that the edge induced function gf:E→{1,2,…,k} is also injective. Let c(f) be the positive integer r such that {1,2,…,r}⊂gf(E) and r+1∉gf(E). The edge consecutive gracefulness ecg(G) is defined by ecg(G)=maxf{c(f)} where the maximum is taken over all injective functions f:V→N∪{0} such that gf is also injective. In this paper, we present several basic results on ecg and use this parameter to prove that any connected graph G can be embedded as an induced subgraph of a graceful graph and as an induced subgraph of an eulerian graceful graph.