Vertex graceful labeling of some classes of graphs

A. P. Santhakumaran, P. Balaganesan
2018 Proyecciones  
A connected graph G = (V, E) of order atleast two, with order p and size q is called vertex-graceful if there exists a bijection f : V → {1, 2, 3, · · · p} such that the induced function f * : is a bijection. The bijection f is called a vertex-graceful labeling of G. A subset S of the set of natural numbers N is called consecutive if S consists of consecutive integers. For any set X, a mapping f : X → N is said to be consecutive if f (X) is consecutive. A vertex-graceful labeling f is said to
more » ... ling f is said to be strong if the function f 1 : E → N defined by f 1 (e) = f (u) + f (v) for all edges e = uv in E forms a consecutive set. It is proved that one vertex union of odd number of copies of isomorphic caterpillars is vertex-graceful and any caterpillar is strong vertex-graceful. It is proved that a spider with even number of legs (paths) of equal length appended to each vertex of an odd cycle is vertex-graceful. It is also proved that the graph lA(m j , n) is vertex-graceful for both n and l odd, 0 ≤ i ≤ n − 1, 1 ≤ j ≤ m i . Further, it is proved that the graph A(m j , n) is strong vertex-graceful for n odd, 0 ≤ i ≤ n − 1, 1 ≤ j ≤ m i .
doi:10.4067/s0716-09172018000100019 fatcat:hhh3aow6vbak3iojgthvvzgiay