Hamilton Connectivity of Convex Polytopes with Applications to Their Detour Index

Abstract

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

Modified Detour Index of Hamiltonian Connected (Laceable) Graphs

“…Therefore, R n is non-bipartite graph. Further, in (10), it is proved that S n is Hamiltonian connected. That is, S n is attainable for every arbitrary pair of vertices.…”

“…For n ≥ 4 by R n we denote the graph of the convex polytope (10) which is obtained as a combination of the graph of a prism and the graph of an antiprism, where V (R n ) = {x j , y j , z j : 1 ≤ j ≤ n} and the edge set of R n is given by: E(R n ) = {(x j , x j+1 ), (x j , y j ), (x j+1 , y j ), (y j , y j+1 ), (y j , z j ), (z j , z j+1 ); 1 ≤ j ≤ n} We make the convention that u n+l = u l , v n+l = v l and w n+l = w l to simplify the notation. See Figure 1 to view the ndimensional convex polytope R n .…”

Hamilton-Connected Mycielski Graphs∗

“…. , y n ∪ z { } with edges zy i for all i and edges x i x j , y i x j , and x i y j for all edges v i v j in G. In recent years, a number of papers are devoted to various properties of Mycielski graphs, such as Hamiltonconnectivity, Hamiltonicity [3][4][5][6][7], total chromatic number [8,9], circular chromatic number [10][11][12][13][14], and connectivity [15,16]. Fisher et al [4] obtained the following results.…”