| Popis: |
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. |
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