Classification of finite groups with toroidal or projective-planar permutability graphs
نویسندگان
چکیده
Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph having all the proper subgroups of G as its vertices, and two subgroups are adjacent in Γ(G) if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of K3,3, K1,5, C6, P5, or P6 as a subgraph.
منابع مشابه
The complement of proper power graphs of finite groups
For a finite group G, the proper power graph P∗(G) of G is the graph whose vertices are non-trivial elements of G and two vertices u and v are adjacent if and only if u 6= v and um = v or vm = u for some positive integer m. In this paper, we consider the complement of P∗(G), denoted by P(G). We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a ...
متن کامل$n$-Array Jacobson graphs
We generalize the notion of Jacobson graphs into $n$-array columns called $n$-array Jacobson graphs and determine their connectivities and diameters. Also, we will study forbidden structures of these graphs and determine when an $n$-array Jacobson graph is planar, outer planar, projective, perfect or domination perfect.
متن کاملThe structure of K3, 3-subdivision-free toroidal graphs
We describe the structure of 2-connected non-planar toroidal graphs with no K3,3-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K5-subdivision in [A. Gagarin and W. Kocay, “Embedding graphs containing K5-subdivisions”...
متن کاملM-degrees of quadrangle-free planar graphs
The M-degree of an edge xy in a graph is the maximum of the degrees of x and y. The M-degree of a graph G is the minimum over M-degrees of its edges. In order to get upper bounds on the game chromatic number, He et al showed that every planar graph G without leaves and 4cycles has M-degree at most 8 and gave an example of such a graph with M-degree 3. This yields upper bounds on the game chroma...
متن کاملRelative non-Normal Graphs of a Subgroup of Finite Groups
Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...
متن کامل