Tree-Antimagicness of Disconnected Graphs

Martin Bača,Zuzana Kimáková,Andrea Semaničová-Feňovčíková,Muhammad Awais Umar

doi:10.1155/2015/504251

Martin Bača, Zuzana Kimáková + Show 2 more

Open Access

https://doi.org/10.1155/2015/504251

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Abstract

A simple graphGadmits anH-covering if every edge inE(G)belongs to a subgraph ofGisomorphic toH. The graphGis said to be (a,d)-H-antimagic if there exists a bijection from the vertex setV(G)and the edge setE(G)onto the set of integers1, 2, …,VG+E(G)such that, for all subgraphsH′ofGisomorphic toH, the sum of labels of all vertices and edges belonging toH′constitute the arithmetic progression with the initial termaand the common differenced.Gis said to be a super (a,d)-H-antimagic if the smallest possible labels appear on the vertices. In this paper, we study super tree-antimagic total labelings of disjoint union of graphs.

Highlights

We consider finite undirected graphs without loops and multiple edges
The vertex and edge set of a graph G are denoted by V(G) and E(G), respectively
An (a, d)-Hantimagic total labeling of graph G admitting an H-covering is a total labeling with the property that, for all subgraphs H󸀠 isomorphic to H, the H󸀠-weights form an arithmetic progression a, a + d, a + 2d, . . . , a + (t − 1)d, where a > 0 and d ≥ 0 are two integers and t is the number of all subgraphs of G isomorphic to H

Summary

Introduction

We consider finite undirected graphs without loops and multiple edges. The vertex and edge set of a graph G are denoted by V(G) and E(G), respectively. A labeling of type (1, 1, 0), that is, a total labeling, of a plane graph is said to be d-antimagic if for every positive integer s the set of weights of all s-sided faces is Ws = {as, as + d, as + 2d, . Second we can see that the edges of ⋃mi=1 Gi under the labeling f use integers from pm + 1 up to (p + q)m It means that if i = 1, the edges of G1 successively assume values [(p + 1)m, (p + 2)m, (p + 3)m, . For the weight of every subgraph Tij isomorphic to the tree T under the labeling f, we have wtf (Tij) = ∑ f (V) + ∑ f (e). The disjoint union of arbitrary number of copies of G, that is, mG, m ≥ 1, admits a super (b, 1)-T-antimagic total labeling. The resulting labeling g is a (b, 1)-T-antimagic total labeling

Disjoint Union of Certain Families of Graphs

Conclusion