Heptagonal graceful labeling of caterpillar and path related graphs

Abstract

Numbers of the form for all n ≥ 1 are called heptagonal numbers. Let G be a graph with p vertices and q edges. Let f∶ V (G)→ {0,1,2,...,Nq} where Nq is the qth heptagonal number be an injective function. Define the function f∶ (E(G)) → {N1, N2, N3,...,Nq} such that f*(uv) = |f(u) - f(v)| for all edges uv ∈ E(G). If f* (E(G)) is a sequence of distinct consecutive numbers {N1, N2, N3,...,Nq} then the function f is said to be heptagonal graceful labeling and the graph which admits such a labeling is called a heptagonal graceful graph.