P. Turán has asked the following question:. Let I12 be the graph determined by the vertices and edges of an icosahedron. What is the maximum number of edges of a graph Gn of n vertices if Gn does not contain I12 as a subgraph? We shall answer this question by proving that if n is sufficiently large, then there exists only one graph having maximum number of edges among the graphs of n vertices and not containing I12. This graph Hn can be defined in the following way:. Let us divide n - 2 vertices into 3 classes each of which contains [ (n-2) 3] or [ (n-2) 3] + 1 vertices. Join two vertices iff they are in different classes. Join two vertices outside of these classes to each other and to every vertex of these three classes.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics