## 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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