Hamiltonian decomposition of Cayley graphs of degree 4

On isomorphic factorizations of circulant graphs

“…If k = 2n, X itself is the isofactor. If k = n, then the main theorem of [3] implies that X has a decomposition into two Hamilton cycles. This takes care of k = 2n and k = n.…”

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.…”

“…For the remainder of this article, E(F ) will be colored red, E(H 1 ) colored blue, and (1,2), (5, 7)} ) (see Figure 7). If B = (3, 1) , then the quotient graph, X, of X by B is HD into (1,2), (5,2), (4,0), (3,6), (5,7), (0, 0) (5,2), (3,6), (1,2), (5, 7), (4,0), (0, 0).…”

“…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]). Every connected k-regular Cayley graph on a finite Abelian group A is Hamilton-decomposable if k ≤ 5 or k = 6 and |A| is odd.…”

Bipartite 2‐Factorizations of Complete Multipartite Graphs

“…, m t ], and exponents may be used to indicate multiple cycles of the same length. For example, [4,4,6,6,6, 10] may be denoted by [4 2 , 6 3 , 10]. Throughout the paper, the meaning of any notation involving an exponent is as defined in this paragraph.…”