Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 359–370 Odd edge-colorability of subcubic graphs * Risto Atanasov Department of Mathematics and Computer Science, Western Carolina University, 28723 Cullowhee, NC, USA Mirko Petruševski Department of Mathematics and Informatics, Faculty of Mechanical Engineering, Sts. Cyril and Methodius University, 1000 Skopje, Republic of Macedonia Riste Škrekovski Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Faculty of Information Studies, 8000 Novo Mesto, Slovenia University of Primorska, FAMNIT, 6000 Koper, Slovenia Received 16 October 2015, accepted 21 January 2016, published online 1 March 2016 Abstract An edge-coloring of a graph G is said to be odd if for each vertex v of G and each color c, the vertex v either uses the color c an odd number of times or does not use it at all. The minimum number of colors needed for an odd edge-coloring of G is the odd chromatic index χ 0 o (G). These notions were introduced by Pyber in [7], who showed that 4 colors suffice for an odd edge-coloring of any simple graph. In this paper, we consider loopless subcubic graphs, and give a complete characterization in terms of the value of their odd chromatic index. Keywords: Subcubic graph, odd edge-coloring, odd chromatic index, odd edge-covering, T -join. Math. Subj. Class.: 05C15 1 Introduction 1.1 Terminology and notation Throughout the article we mainly follow the terminology and notation used in [1, 11]. A graph G =(V (G),E(G)) is always regarded as being finite, i.e. having a finite nonempty * This work is partially supported by ARRS Program P1-0383. E-mail addresses: ratanasov@email.wcu.edu (Risto Atanasov), mirko.petrushevski@gmail.com (Mirko Petruševski ), skrekovski@gmail.com (Riste Škrekovski) cb This work is licensed under http://creativecommons.org/licenses/by/3.0/