2016/09/06 by Dhruv Rohatgi, Rohatgi, Dhruv
Computer Science · #Advanced Graph Theory Research #Algorithms and Data Compression #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1609.01367
openalex publication_date 2016/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Trotter and Erdös found conditions for when a directed m × n grid graph on a torus is Hamiltonian. We consider the analogous graphs on a two-holed torus, and study their Hamiltonicity. We find an O(n4) algorithm to determine the Hamiltonicity of one of these graphs and an O(log(n)) algorithm to find the number of diagonals, which are sets of vertices that force the directions of edges in any Hamiltonian cycle. We also show that there is a periodicity pattern in the graphs' Hamiltonicities if one of the sides of the grid is fixed; and we completely classify which graphs are Hamiltonian in the cases where n=m, n=2, the m × n graph has 1 diagonal, or the (m)/(2) × (n)/(2) graph has 1 diagonal.