On Graphs with Adjacent Vertices of Large Degree
نویسنده
چکیده
Let g(n, m) denote the class of simple graphs on n vertices and m edges and let C E 9( a, m). For suitably restricted values of m, C will necessarily contain certain prescribed subgraphs such as cycles of given lengths and complete graphs . For example> if m > 7nZ then G contains cycles of all lengths up to Li(n+ 3) J . Recently we have established a number of results concerning the existence of certain subgraphs (cliques and cycles) in the subgraph of G induced by the vertices of C having some prescribed minimum degree . In this paper, we present some further results of this type. In particular, we prove diet every G E 9(+a m) contains a pair of adjacent vertices each having degree (in C) at least f(n,m) and determine the best possible value o(/(n, en), Form > rn2 we find that C contains a triangle with a pair of vertices satisfying this same degree restriction . Some open problems are discussed .
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تاریخ انتشار 2005