Edge Disjoint Hamilton Cycles in Knödel Graphs

All graphs considered here are simple and finite unless otherwise stated. LetCk (resp.Pk) denote the cycle (resp. path) on k vertices. For a graphG, if its edge set E(G) can be partitioned into E1,E2, . . . ,Ek such that 〈Ei〉 ∼= H, for all i, 1 ≤ i ≤ k, then we say thatH decomposesG.A k-factor ofG is a k-regular spanning subgraph of it. A k-factorization of a graphG is a partition of the edge set ofG into E1, E2, . . . , Es such that 〈Ei〉, 1 ≤ i ≤ s, is a k-factor. We say that a k-regular graphG admits a Hamilton cycle decomposition, if the edge set of G can be partitioned into Hamilton cycles or Hamilton cycles together with a 1-factor according as k is even or odd, respectively. If H1, H2, . . . , Hk are edge disjoint subgraphs of G such that ⋃k i=1 Hi = G, then we write G = H1 ⊕H2 ⊕ . . .⊕Hk. The complete graph on n vertices is denoted by Kn. Let G be a bipartite graph with bipartition (X,Y ), where X = {x0, x1, . . . , xn−1}, Y = {y0, y1, . . . , yn−1}; the edge xiyi+l is called an edge of jump l from X to Y in G, where addition is taken modulo n; the same edge is called an edge of jump n − l from Y to X. If G contains the edges Fl(X,Y ) = {xiyi+l|0 ≤ i ≤ n − 1, where addition in the subscript is taken modulo n}, 0 ≤ l ≤ n − 1, then we say that G has the 1-factor of jump l from X to Y. Clearly, if G = Kn,n, then E(G) = ⋃n−1 i=0 Fi(X,Y ). Note that Fi(X,Y ) = Fn−i(Y,X), 0 ≤ i ≤ n − 1, where we assume Fn(X,Y ) = F0(X,Y ) = F0(Y, X). An anti-directed path P is a digraph, whose underlying graph is a path, in which any two consecutive arcs of P are either directed toward or away from the common incident vertex in P. Similarly, we define anti-directed cycles, see Figures 1(a) and 1(b). A digraph ~ G = (V, A) is denoted by ~ G. A digraph