Hamiltonian cycles and paths in Cayley graphs and digraphs — A survey

Gray code numbers for graphs

“…It is known that Cay(X : S n ) is connected if and only if X is a generating set. It is an open question as to whether every connected Cayley graph on S n is Hamiltonian [6]. This question has been settled by Kompel'makher and Liskovets in the case where the generating set X is a set of transpositions [11].…”

Hamilton decompositions of 6-regular Cayley graphs on even Abelian groups with involution-free connections sets

“…Trivially, the path P = 0, 1, 2, 3, 4, 5, 6, 7 forms a P 8 2,6 -path, and Theorem 5.9 produces the result. If {x, y} = {1, 3}, then 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 and H 2 := 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0 is a Hamilton decomposition, and P = 0, 3,4,5,6,7,8,9 forms a P 8 2,6 -path. Similarly, if {x, y} = {1, 2}, then 2, 1, 3, 4, 5, 6, 7, 8, 9, 0 and H 2 := 7, 5, 3, 2, 4, 6, 8, 0, 1, 9, 7 is a Hamilton decomposition, and P = 0, 9, 7, 5, 3, 2, 4, 6 forms a P 8 2,6 -path.…”

“…If m ≥ 6 is even, n ≥ 9 is odd, and exactly one of t 1 and t 2 is even, then X is HD. (4,5), (0, 6), (4,7), (0,8), (4,9), (0, 10), (1,0), (4,1), (0, 2) = a 1 , a 2 , . .…”

“…Computational Appendix H2:0,2,17,15,5,7,9,11,21,19,10,12,3,1,23,8,18,20,22,13,4,6,16,14 Computational Appendix (0,7), (1,10), (0,1), (0,9) Cay(Z2⊕Z12;{ (1,3), (1,5),(0,3)} ) H0:(0,4),(0,1), (1,4), (1,7),(0,2), (1,5), (0,8), (1,11), (1,2), (0,5), (1,8), (0,11), (1,6), (1,3), (0,6), (0,9), (1,0),(0,7),(0,10), (1,1), (1,10), (0,3),(0,0), (1,9) H1:(0,7), (1,2), (0,9), (1,6), (0,3), (0,…”

“…In fact, connected Cayley graphs of Abelian groups are rich with Hamilton cycles (see ) and Alspach [2] conjectured in 1984 that this class of graphs is Hamilton-decomposable. Alspach's conjecture remains open in general (see the survey by Curran-Gallian [9]) despite being investigated from many vantage points: the valency, the group order, the group type, restrictions on the connection set, etc. The following theorem is a summary of complete results when only the valency is considered: Theorem 1.1 (Alspach et al [3], Bermond et al [5], Dean [11], Fan et al [12], Liu [14,15], Westlund et al [19]).…”