Directed Hamilton Cycle Decompositions of the Tensor Products of Symmetric Digraphs

Abstract

A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. Manikandan and Paulraja conjectured that if G and H are Hamilton cycle decomposable circulant graphs with at least one of them is nonbipartite, then their tensor product is Hamilton cycle decomposable. In this paper, we have proved that, if G is a Hamilton cycle decomposable circulant graph with certain properties and H is a Hamilton cycle decomposable multigraph, then their tensor product is Hamilton cycle decomposable. In particular, tensor products of certain sparse Hamilton cycle decomposable circulant graphs are Hamilton cycle decomposable. * A circulant X = Circ(n; L) is a graph with vertex set V(X) = {u 0 , u 1 , . . . , u n−1 } and edge set E(X) = {u i u i+ℓ | i ∈ Z n , ℓ ∈ L}, where L ⊆ 1, 2, . . . , ⌊ n 2 ⌋ and Z n is the set of integers modulo n. The elements of L are called jumps. Clearly, every circulant graph of order n is a Cayley graph with the underlying group being Z n .The problem of finding Hamilton cycle decompositions of product graphs is not new. Hamilton cycle decompositions of various product graphs, including digraphs, have been studied by many authors; see, for example, [1,6,[9][10][11][13][14][15][16][17][18]. It has been conjectured [6] that if both G and H are Hamilton cycle decomposable graphs, then G H is Hamilton cycle decomposable, where denotes the cartesian product of graphs. This conjecture has been verified to be true for a large classes of graphs [18]. Baranyai and Szász [5] proved that if both G and H are even regular Hamilton cycle decomposable graphs, then G • H is Hamilton cycle decomposable. In [16], Ng has obtained a partial solution to the following conjecture of Alspach et al. [1]: If D 1 and D 2 are directed Hamilton cycle decomposable digraphs, then D 1 • D 2 is directed Hamilton cycle decomposable. Jha [10] conjectured the following Conjecture 1.1 [10] If both G and H are Hamilton cycle decomposable graphs and G × H is connected, then G × H is Hamilton cycle decomposable. Conjecture 1.1 was disproved, see [3]. Because of this, finding Hamilton cycle decompositions of the tensor products of Hamilton cycle decomposable graphs is considered to be difficult. Though Conjecture 1.1 has been disproved, we believe that if the graphs G and H are suitably chosen, that is, with some suitable conditions imposed on them, then G × H may have Hamilton cycle decomposition. In [2] and [13] it has been proved that K r × K s and K r,r × K m are Hamilton cycle decomposable; in [14] it is shown that the tensor product of two regular complete multipartite graphs is Hamilton cycle decomposable. Hamilton cycle decompositions of the tensor products of complete bipartite graphs and complete multipartite graphs are dealt with in [15]. Also in [17], Paulraja and Sivasankar proved that are directed Hamilton cycle decomposable. It can be observed that K r , K r, r , K r • K s are circulant graphs. Based on the results of [10, 11, 13-15], Manikandan and Paulraja conjectured the following: Conjecture 1...

On the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform‐length cycles

“…The above corollary shows how to “blow up the holes” in a $${\overrightarrow{C}}_{t}$$‐factorizaton by either keeping the cycle length, or “blowing up” the cycle length by the same odd factor. Note that Statement (b) also follows from [19, Lemma 2.11], and Statement (a) can be obtained from [20, Corollary 5.7] by appropriately orienting each cycle.…”

Decomposition of the tensor product of complete graphs into cycles of lengths 3 and 6

“…Moreover, a resolvable 2k-cycle decomposition of K m × K n and a decomposition of K m × K n into closed trails of length k have been studied in [33,34]. Besides that, Hamilton cycle decompositions of the graphs [8,[28][29][30][31]35]. Hence K m × K n is proved to be an important proper spanning subgraph of the regular complete…”

On hamiltonian decompositions of tensor products of graphs

“…The problem of finding hamiltonian decompositions of product graphs is not new. hamiltonian decompositions of various product graphs, including digraphs, have been studied by many authors; see, for example, [1,6,9,10,11,13,14,15,16,18,17]. It has been conjectured [6] that if both G and H are hamiltonian decomposable graphs, then G H is hamiltonian decomposable, where denotes the cartesian product of graphs.…”

“…Hamiltonian decompositions of the tensor products of complete bipartite graphs and complete multipartite graphs are dealt with in [15]. Also in [17], Paulraja and Sivasankar proved that (K r × K s ) * , ((K r • K s ) × K n ) * , ((K r × K s ) × K m ) * , ((K r • K s ) × (K m • K n )) * and (K r,r × (K m • K n )) * are directed hamiltonian decomposable. It can be observed that K r , K r, r , K r •K s are circulant graphs.…”