Discussiones Mathematicae Graph Theory 31(1) (2011) 79-113
doi: 10.7151/dmgt.1531

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A MAGICAL APPROACH TO SOME LABELING CONJECTURES

Ramon M. Figueroa-Centeno

Mathematics Department, University of Hawai'i at Hilo
200 W. Kawili St., Hilo, HI 96720, USA
e-mail: ramonf@hawaii.edu

Rikio Ichishima

College of Humanities and Sciences, Nihon University
3-25-40 Sakurajosui Setagaya-ku, Tokyo 156-8550, Japan
e-mail: ichishim@chs.nihon-u.ac.jp

Francesc A. Muntaner-Batle

Graph Theory and Applications Research Group
School of Electrical Engineering and Computer Science
Faculty of Engineering and Built Environment
University of Newcastle, NSW 2308, Australia
e-mail: famb1es@yahoo.es

Akito Oshima

Department of Mathematical Information Science
Faculty of Science, Tokyo University of Science
1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
e-mail: akito_o@rs.kagu.tus.ac.jp

Abstract

In this paper, a complete characterization of the (super) edge-magic linear forests with two components is provided. In the process of establishing this characterization, the super edge-magic, harmonious, sequential and felicitous properties of certain 2-regular graphs are investigated, and several results on super edge-magic and felicitous labelings of unions of cycles and paths are presented. These labelings resolve one conjecture on harmonious graphs as a corollary, and make headway towards the resolution of others. They also provide the basis for some new conjectures (and a weaker form of an old one) on labelings of 2-regular graphs.

Keywords: edge-magic labelling, edge-magic total labelling, felicitous labelling, harmonious labelling, sequential labelling.

2010 Mathematics Subject Classification: 05C78.

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Received 21 September 2009
Revised 6 April 2010
Accepted 6 April 2010