Minimum Degree, Factors and Magic Graphs
نویسنده
چکیده
The main result of this paper is as follows: Let G be a graph and m a positive integer such that (i) |V (G)| ≥ 4m+1, (ii) δ(G) ≥ |V (G)| 2 +1. For any subset {e1, . . . , em+1} of E(G), the graph G − {e1, . . . , em} possesses a 2-factor containing em+1. The above Theorem also yields a Dirac type sufficient condition for a graph to be magic. All graphs considered are assumed to be simple and finite. We refer the reader to [1] for standard graph theoretic terms not defined in this paper. Let G be a graph. The degree dG(u) of a vertex u in G is the number of edges of G incident with u. The minimum degree of a vertex in G is denoted by δ(G). If X and Y are disjoint subsets of V (G), we will write EG(X,Y ) and eG(X,Y ) for the set and the number respectively of the edges of G joining X to Y . The number of connected components of G is denoted by ω(G). An edge e of G is said to be subdivided when it is deleted and replaced by a path of length two connecting its ends, the internal vertex of this path being a new vertex. For any set S of vertices in G, we define the neighbour set of S in G to be the set of all vertices adjacent to vertices in S; this set is denotd by NG(S).
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