Directed edge-graceful labeling of digraph consisting of c cycles of the same size

Abstract

A graph G = (V, E) is said to be directed edge-graceful if there exists an orientation of G and a directed edge-graceful labeling f of the arcs of G with {1, 2, 3, … ,|E|} such that the induced mapping g on V defined by g(v) = (f⁺(v) - f⁻(v)) (mod |V|) is a bijection, where f⁺(v) is the sum of the labels f of all arcs with head v and f⁻(v) is the sum of the labels f of all arcs with v as tail. This article consider the graph C(c × a) which obtained by identifying a vertex of c cycles C_a to a single vertex. Then, we give an orientation to each edge of C(c × a) and construct a directed edge labeling to this digraph, where (i) a is an odd integer such that a ≥ 3 and c is an integer such that c ≥ 2 and (ii) a and c are even integers such that a ≥ 4 and c ≥ 2. Finally, we prove that the constructed edge labelings are directed edge-graceful labelings.

Subject Classification:

• 05C78

Keywords:

• Directed edge-graceful labeling
• Digraph labeling
• Cycle