Graphs with unavoidable subgraphs with large degrees

Let `,(n, m) denote the class of simple graphs on n vertices and m edges and let G C `6(n, m) . There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc . For example, Turan's theorem gives a sufficient condition for G to contain a K,,„ in terms of the number of edges in G . In this paper we prove that, for m = an t , a > (k 1)/2k, G contains a K,,,, each vertex of which has degree at least f(a)n and determine the best possible f(a) . For m = Ln 2 /4J + 1 we establish that G contains cycles whose vertices have certain minimum degrees . Further, for m an t, a > 0 we establish that G contains a subgraph Hwith S(H)_ f(a, n) and determine the best possible value of f(a, n) . *This research was conducted, in part, while visiting the Department of Combinatorics and Optimization, University of Waterloo . 'This research was conducted, in part, while visiting the Department of Statistics and Actuarial Science, University of Waterloo . Journal of Graph Theory, Vol . 12, No . 1, 17-27 (1988) 1988 by John Wiley & Sons, Inc . CCC 0364-9024/881010017-11504 .00 18 JOURNAL OF GRAPH THEORY