Comparison of sufficient degree based conditions for Hamiltonian graph | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2019. № 45. DOI: 10.17223/20710410/45/6

A graph G is said to be Hamiltonian if it contains a spanning cycle, i.e. a cycle that passes through all of its vertices. The Hamiltonian cycle problem is NP-complete, and many sufficient conditions have been found after the first sufficient condition proposed by Dirac in 1952. In this paper for all graphs with a number of vertices up to 12, the most popular sufficient degree based conditions for Hamiltonian graph are compared: theorems by Dirac, Ore, Posa, Chvatal and Bondy-Chvatal. The number of graphs which satisfy each condition is counted. With the number of vertices from 3 to 12, the number of graphs satisfying the Dirac condition is 1, 3, 3, 19, 29, 424, 1165, 108376, 868311, 495369040; the number of graphs satisfying the Ore condition is 1, 3, 5, 21, 68, 503, 4942, 128361, 5315783, 575886211; the number of graphs satisfying the Posha condition is 1, 3, 6, 31, 190, 2484, 53492, 2683649, 216082075, 40913881116; the number of graphs satisfying the Chvatal condition is 1, 3, 6, 34, 194, 2733, 54435, 2914167, 218674224, 43257613552 and the number of graphs satisfying the Bondy - Chvatal condition is 1, 3, 7, 45, 352, 5540, 157016, 8298805, 802944311, 141613919605. This result is the best one: about 90 % of the Hamiltonian graphs satisfy condition proposed by Bondy and Chvatal in 1976. The FHCP Challenge Set is a collection of 1001 instances of the Hamiltonian Cycle Problem, ranging in size from 66 vertices up to 9528. All graphs from the FHCP Challenge Set were checked whether they satisfy considered conditions. It turned out that 11 graphs satisfy the Bondy - Chvatal condition: no. 59 (with 400 vertices), no. 72 (460), no. 79 (480), no. 84 (500), no.90 (510), no.96 (540), no. 128 (677), no. 134 (724), no. 150 (823), no. 162 (909), and no. 188 (with 1123 vertices). For these graphs we can check and find Hamiltonian cycle using Bondy - Chvatal's theorem with computational complexity O(n⁴) where n is the number of graph vertices.