11/11/2022 0 Comments Number of edges in hypercube![]() It has 8 verticies and 12 edges and 6 faces.Ī 4d hypercube has 16 verticies and not sure how many edges or 3d faces. Again not sure how many faces it has?Ī 3d hypercube is a cube. Not sure how many faces it has?Ī 2d hypercube is a square. Suppose that the result holds for Qn−1, for some n ≥ 4.So I guess a 1d hypercube is a line segment. By Lemma 6, the statement holds for n = 3. We prove the statement by induction on n. Proof: It is clear that the result holds for Q2. Then there exist n − 1 mutually fully independent paths Pl Independent edges of Qn with n ≥ 2, ei = (bi, wi), In ith dimension and the edge (u, v) is called an ith dimension edge. Position, then the edge between them is said to be If the binary labels of u and v differ in ith Vertices if and only if their binary labels differ inĮxactly one bit position. Is a graph with 2 n vertices, and each vertex u can be distinctly labeled by an n-bit binary string, If every two cycles are independent with respect toĪn n-dimensional hypercube, denoted by Qn, Mutually independent with respect to the edge (x, y) , vm, v1 i passing through an edge (x, y)Īre independent with respect to the edge (x, y), if u1 = v1 = x, um = vm = y and ui 6= vi for 2 ≤ i ≤ m − 1.Indepen-dent if each pair of them are fully indepenindepen-dent. Two paths P1 and P2 are fully independent if if every two different paths are independent. , Pnfrom a to b are mutually independent., vmi from a to b are independent if u1 = v1 = a, um = vm = b, and ui 6= vi for 2 ≤ i ≤ m − 1.We now introduce a relatively new concept. A bipartite graph is bipanconnected if, for any two different vertices x and y, there exists a path of length l joining x and y, for every l,ĭG(x, y) ≤ l ≤ |V (G)| − 1 and (l − dG(x, y))īeing even. Therefore, the concept of bipanconnected graphs is ![]() Graph with at least 3 vertices is not panconnected. Of panconnected graphs is proposed by Alavi and A graph is panconnected if, for any two different vertices x and y, there exists a path of length l joining x and y, for every l, dG(x, y) ≤ l ≤ |V (G)| − 1. A bipartite graph is edge-bipancyclic if each edge lies on cy-cles of every even length from 4 to |V (G)|. A graph G is called edge-pancyclic if each edge lies on cycles A graph G is pancyclic if G includes cycles of all lengths. Aīipartite graph G is hamiltonian laceable if thereĮxists a hamiltonian path joining any two verticesįrom different partite sets. Least three vertices is not hamiltonian connected. It is easy to see that any bipartite graph with at A graph G = (B ∪ W, E) isīipartite if V (G) is the union of two disjoint sets B and W such that every edge joins B with W. If there exists a hamiltonian path between any twoĭifferent vertices of G. The length of aĪ path is a hamitonian path if it contains all the vertices of G.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |