Let ð�’¢(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ð�’¢ (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 Kk + 1 in terms of the number of edges in G. In this paper we prove that, for m = αn2, α > (k ‐ 1)/2k, G contains a Kk + 1, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ⌊n2/4⌋ + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn2, α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).
ASJC Scopus subject areas
- Geometry and Topology