TY  - JOUR
AU  - Hayat, Sakander
AU  - Imanda, Bagus
AU  - Khan, Asad
AU  - Alenazi, Mohammed J. F.
PY  - 2025
TI  - Three infinite families of Hamilton-connected convex polytopes and their detour index
JO  - AIMS Mathematics
SP  - 12343
EP  - 12387
VL  - 10
IS  - 5
AB  - A path in a graph encompassing its whole vertex set is called Hamiltonian. Such a path with sharing the same initial and terminal vertices is called a Hamiltonian cycle. A graph comprising a Hamiltonian path (resp. cycle) is said to be traceable (resp. Hamiltonian). Graphs possessing Hamiltonian paths between every pair of their vertices are said to be Hamilton-connected. The computational complexity of evaluating a graph to be Hamilton-connected is NP-complete. A detour is the longest path in a graph. The detour index is the sum of the length of detours between every unordered pair of vertices. Computing the detour index of a graph is an NP-complete problem as well. A finite subset    P  ⊂            R        ε   is called a convex polytope if    P is a convex hull. In this paper, we devised two distinct methods to prove a graph to be Hamilton-connected and employed these methods to construct some infinite families of Hamilton-connected convex polytopes. The convex polytope        B    ε   has been shown to be non-Hamilton-connected in the literature. We showed that the existing proof for        B    ε   is false and showed that this family is, in fact, Hamilton-connected. The paper is concluded with study implications followed by some future directions.
UR  - https://doi.org/10.3934/math.2025559
DO  - 10.3934/math.2025559