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Face Antimagic Labelings for a Special Class of Plane Graphs Cab

Martin Baéa1, Edy Tri Baskoro2, Yus M. Cholily2
1Department of Appl. Mathematics Technical University, Letnd 9, 042 00 Koiice, Slovak Republic
2Department of Mathematics Institut Teknologi Bandung Jl. Ganesa 10 Bandung 40132, Indonesia

Abstract

Suppose G=(V,E,F) is a finite plane graph with vertex set V(G), edge set E(G), and face set F(G). A bijection λ:V(G)E(G)F(G){1,2,3,,|V(G)|+|E(G)|+|F(G)|} is called a labeling of type (1,1,1). The weight of a face under a labeling is the sum of the labels (if present) carried by that face and the edges and vertices surrounding it. A labeling of a plane graph G is called d-\emph{antimagic} if for every number s3, the set of s-sided face weights is

Ws={as+id:0ifs}

for some integers as and d (a>0, d0), where fs is the number of s-sided faces. We allow different sets Ws for different s.

In this paper, we deal with d-\emph{antimagic} labelings of type (1,1,1) for a special class of plane graphs Cab and we show that a Cab graph has d-antimagic labeling for d{a2,a1,a+1,a+2}.