Further results on edge even graceful labeling of the join of two graphs

Abstract

Abstract In this paper, we investigated the edge even graceful labeling property of the join of two graphs. A function f is called an edge even graceful labeling of a graph G=(V(G),E(G)) with p=|V(G)| vertices and q=|E(G)| edges if f:E(G)→{2,4,...,2q} is bijective and the induced function f ∗:V(G) →{0,2,4,⋯,2q−2 }, defined as f ∗ ( x ) = ( ∑ xy ∈ E ( G ) f ( xy ) ) mod ( 2 k ) $ f^{\ast }(x) = ({\sum \nolimits }_{xy \in E(G)} f(xy)~)~\mbox{{mod}}~(2k) $ , where k=m a x(p,q), is an injective function. Sufficient conditions for the complete bipartite graph K m,n =m K 1+n K 1 to have an edge even graceful labeling are established. Also, we introduced an edge even graceful labeling of the join of the graph K 1 with the star graph K 1,n , the wheel graph W n and the sunflower graph s f n for all n ∈ ℕ $n \in \mathbb {N}$ . Finally, we proved that the join of the graph K ¯ 2 $\overline {K}_{2}~$ with the star graph K 1,n , the wheel graph W n and the cyclic graph C n are edge even graceful graphs.

Keywords

• Complete bipartite graph
• Wheel graph
• Sunflower graph
• Edge even graceful labeling
• Join of two graphs