A note on the Turán number of disjoint union of wheels

Abstract

The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. A wheel Wn is an n-vertex graph formed by connecting a single vertex to all vertices of a cycle Cn−1. Let mW2k+1 (k≥3) denote the graph defined by taking m vertex disjoint copies of W2k+1. For sufficiently large n, we determine the Turán number and all extremal graphs for mW2k+1 (k≥3). Let Wh be the family of graphs obtain by the disjoint union of a finite number of wheels, such that, the number of even wheels in the union is h, (h≥1). For any W∈Wh, we also provide the Turán number and all extremal graphs for W, when n is sufficiently large.