A c-matching is simply a pair of disjoint edges. A packing P of c-matchings into a graph H can be thought of as an edge coloring of H in which each color class has exactly c edges and some edges may be left uncolored. The (vertex) intersection graph I(P) of a packing P has a vertex for each matching and an edge between two vertices iff the corresponding matchings share an edge. Thus when c=l, this is the familiar line graph L(H). In this paper we examine which types of graphs can be represented as two matchings. Specifically, we restrict our analysis of the matching in cubic graphs.
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