Introduction

Let WFGn be the class of graphs on n vertices with the property that the neighborhood of any vertex is acyclic. A graph G is given by its vertex set V(G) and edge set E(G). The subgraph induced by X ⊂ V(G) is denoted by G[X]. The neighborhood N(v) of vertex v is the set of vertices adjacent to v. Note that v ∉ N(v). The degree of v ∈ V(G), denoted by dv or dv(G), is the size of the neighborhood: dv = |N(v)|. The maximum (minimum) degree is denoted by Δ (δ), or Δ(G) (δ(G), respectively) to avoid misunderstandings. A matching M ⊂ E(G) is a set of pairwise disjoint edges. A wheel Wi is obtained from a cycle Ci by adding a new vertex and edges joining it to all the vertices of the cycle; the new edges are called the spokes of the wheel (i ≥ 3, W3 = K4). Therefore, WFGn consists of all graphs on n vertices containing no wheel.