Hindawi Publishing Corporation ISRN Combinatorics Volume 2013, Article ID 358527, 6 pages http://dx.doi.org/10.1155/2013/358527 Research Article Graphs Whose Edge Set Can Be Partitioned into Maximum Matchings Niraj Khare Department of Mathematics, e Ohio State University, Columbus, OH 43210, USA Correspondence should be addressed to Niraj Khare; khare.9@osu.edu Received 17 October 2012; Accepted 19 November 2012 Academic Editors: A. V. Kelarev and B. Wu Copyright © 2013 Niraj Khare. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper provides structural characterization of simple graphs whose edge set can be partitioned into maximum matchings. We use Vizing’s classi�cation of simple graphs based on edge chromatic index. 1. Introduction By a simple graph, we shall mean a graph with no loop and no multiple edges. We will only consider simple graphs with no isolated vertex. We �rst �x some notations. For a graph , , and  would denote the edge set and the vertex set of , respectively. Δ, , and  ′  would denote the maximum degree of any vertex in , the size of a maximum matching in  and the edge chromatic index of , respectively. For   , deg   would denote the degree of the vertex  and  would denote the induced subgraph on   . We now consider simple graphs whose edge set can be partitioned into maximum matchings. Complete graphs and even cycles are some of the examples but there are numerous other examples too. For instance, consider the graph in Figure 1. Vizing’s celebrated theorem states that  ′   Δ   for a simple graph . e de�nition of the edge chromatic index implies that ′  Δ. erefore Vizing classi�ed simple graphs as follows: a simple graph  is in class I if and only if  ′   Δ and a simple graph  is in class II if and only if  ′   Δ  . ere is no structural characterization yet known for the graphs in class I or in class II. It is NP-complete to determine whether a simple graph is in Class I or Class II (see [1]). But under certain restrictions structural characterization of class I and class II graphs has been achieved. It is also known that all planar graphs with maximum degree at least seven are in Class I (see [2, 3]). Another interesting result concerns itself with relative cardinality of Class I and Class II (see [4]). We will characterize Class I and class II graphs whose edge set can be partitioned into maximum matchings. 2. Results Our main aim in this paper is to prove the following results. eorem 1. Let  be a simple graph such that , Δ  , and ||  Δ  ΔΔ.  is a unique graph up to isomorphism if and only if Δ divides . eorem 2. If  is a Class II graph and  has a partition into maximum matchings, then Δ is even and  is the graph with exactly Δ components each isomorphic to  Δ , the complete graph of order Δ  . eorem 3. If  is a Class I graph and  can be partitioned into maximum matchings, then  has a partition into subgraphs that are either  ,Δ or a factor critical graph  such that  can also be partitioned into maximum matchings and  ′    ′ . 3. Preliminaries We �rst establish some basic results that will be extremely useful in the next section. We will be borrowing some ideas and results discussed in [5]. e following de�nition is in [5].