Edge-Oblique Polyhedral Graphs

Extended Abstract denote the set of polyhedral graphs, and let G = G(V, E, F) ∈.The degree d(x) of a vertex × ∈ V(G) is the number of edges incident with x. The degree d(α) of a face α ∈ F(G) is the number of edges incident with α. e = (x,y; α,β) ∈ E(G) denotes an edge incident with the two vertices x, y ∈ V(G), d(x) ≤ d(y) and incident with the two faces α, β ∈ F(G), d(α) ≤ d(β). [K = d(x) ≤ d(y); M = d(α),N = d(β)] is the type of e = (x,y;α,β). S.Jendrol & M.Tkac. [1,2] described all polyhedral graphs having only one or exactly two types of edges. Δ(G) := max{d(a) : a ∈ V ∪ F} is the maximum degree of G. Because G is a polyhedral graph there is no edge of type (3, 3; 3, 3} in G except G is the tetrahedron. 9. High Tatra Conference on Colourings and Cycles in 2000 P.Owens asked the following question: l be two integers with 1 ≤ l ≤ k. Does there exist a polyhedral graph G with k = ∣E(G)∣ edges and l different types of edges? ses l ∈ {1,2} are solved in [2]. In this paper we are interested in the case k = l. G is called to be edge –; oblique if for any type of edges there is at most one edge in E(G) having this type