For k ≥ 2, any graph G with n vertices and (k-1)(n-k+2)+( 2 k-2) edges has a subrgraph of minimum degree at least k; however, this subgraph need not be proper. It is shown that if G has at least (k-1)(n-k+2)+( 2 k-2)+1 edges, then there is a subgraph H of minimal degree k that has at most n - √n ;√6k 3 vertices. Also, conditions that insure the existense of smaller subgraphs of minimum degree k are given.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics