Vertex graceful labeling of some classes of graphs
Abstract: 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 ʄ : V → { 1, 2, 3, ··· p } such that the induced function ʄ*: E → { 0, 1, 2, ··· q-1} defined by ʄ*(uv) = (@...
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| Autores principales: | , |
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| Lenguaje: | English |
| Publicado: |
Universidad Católica del Norte, Departamento de Matemáticas
2018
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| Materias: | |
| Acceso en línea: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172018000100019 |
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| Sumario: | Abstract: 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 ʄ : V → { 1, 2, 3, ··· p } such that the induced function ʄ*: E → { 0, 1, 2, ··· q-1} defined by ʄ*(uv) = (ʄ(u)+ ʄ(v)) (mod q) is a bijection. The bijection ʄ 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 ʄ : X → N $ is said to be consecutive if ʄ(X) is consecutive. A vertex-graceful labeling ʄ is said to be strong if the function ʄ1: E → N defined by ʄ1(e)= ʄ(u) + ʄ(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(mj,n) is vertex-graceful for both n and l odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi. Further, it is proved that the graph A(mj, n) is strong vertex-graceful for n odd, 0 ≤ i ≤ n-1, 1 ≤ j ≤ mi. |
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