Extremal graphs for odd wheels

For a graph H, the Turan number of H, denoted by ex(n,H), is the maximum number of edges of an n-vertex H-free graph. Let g(n,H) denote the maximum number of edges not contained in any monochromatic copy of H in a 2-edge-coloring of Kn. A wheel Wm is a graph formed by connecting a single vertex to all vertices of a cycle of length m-1. The Turan number of W2k was determined by Simonovits in 1960s. In this paper, we determine ex(n,W2k+1) when n is sufficiently large. We also show that, for sufficient large n, g(n,W2k+1)=ex(n,W2k+1) which confirms a conjecture posed by Keevash and Sudakov for odd wheels.

Automated summary

This paper studies the Turan number of graphs and monochromatic copies in 2-edge-coloring, deriving conclusions regarding odd cycles.