Deﬁnitions Core of a graph G is the sub-graph induced of vertices of maximum degree, denoted by G Δ A subset M of E(G) is a matching in the graph G if no two edges in M are adjacent. A 1-factor of a graph G (also called a perfect matching) is a spanning subgraph of G in which every vertex has degree 1. A near 1-factor is a subgraph of G in which every vertex has degree 1 and it spans |V (G)|- 1 vertices. If the edge set of a graph G can be represented as the edge-disjoint union of factors F 1 ,F 2 ,...,F k , we shall refer to {F 1 ,F 2 ,...,F k } as a factorization of graph G. A G-decomposition of H is a partition of E(G) (the edge set of H) that has the property that the subgraph induced by each element of the partition is isomorphic to G. 1

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Core of a graph G is the sub-graph induced of vertices of maximum degree,denoted by G∆

A subset M of E(G) is a matching in the graph G if no two edges in M areadjacent. A 1-factor of a graph G (also called a perfect matching) is a spanningsubgraph of G in which every vertex has degree 1. A near 1-factor is a subgraphof G in which every vertex has degree 1 and it spans |V (G)| − 1 vertices. If theedge set of a graph G can be represented as the edge-disjoint union of factorsF1, F2, . . . , Fk, we shall refer to {F1, F2, . . . , Fk} as a factorization of graph G.

A G-decomposition of H is a partition of E(G) (the edge set of H) thathas the property that the subgraph induced by each element of the partition isisomorphic to G.