Bounding Fitting Heights Of Several Classes Of Character Degree Graphs

By a definition of Lewis in [3], a degree graphΔhas bounded Fitting height if there is a bound on the Fitting height for the solvable group G withΔ(G) =Δ. Lewis proved in [3]: a degree graphΔwith n vertices has bounded Fitting height if and only ifΔhas at most one vertex of degree n -1. Lewis also obtained in [3] that ifΔhas bounded Fitting height, the bound is linear in the number of vertices of the graph. He observed that no graph with bounded Fitting height is found for a solvable group where the bound is bigger than 4. Thus M.L.Lewis has a conjecture(conjecture 5.5 of [5]) that if G be a solvable group whereΔ(G) is a graph with bounded Fitting height. Then G has Fitting height at most 4. Lewis has proved that the conjecture is right at least in the following situations:Theorem A. Let G be a solvable group and suppose thatρ(G) =Ï€₁∪π₂∪p is a disjoint union where |Ï€_i|≥1 for i=1,2. Assume that no prime inÏ€₁ is adjacent inΔ(G) to any prime inÏ€₂. then G has Fitting height at most 4.Theorem B. Let G be a solvable group. Suppose thatΔ(G) is the graph having four vertices where every vertex has degree 2. Then the Fitting height of G is at most 4.In this paper, we proved that the conjecture is right in other situations, and some main theorems as following.Theorem 2.2. Let G be a solvable group with |ρ(G)|≥4. If the total sum of degrees of each derived subgraph with four vertices inΔ(G) is not more than 8, then the Fitting height of G is at most 4. Theorem 3.4. Let G be a solvable group. If there is a fif-order circle inΔ(G), and the total sum of degrees of each derived subgraph with a fif-order circle inΔ(G) is 10, then the Fitting height of G is at most 4.Theorem 4.1. Let G be a solvable group and suppose thatρ(G) =Ï€₁∪π₂ is a disjoint union where |Ï€_i|≥2, p_i,q_i∈π_i, for i = 1,2. Assume that no prime inÏ€₁ is adjacent inΔ(G) to any prime inÏ€₂, except for p₁p₂ and q₁q₂. Then the Fitting height of G is at most 4.