On Hamilton Decompositions of Infinite Circulant Graphs

The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected $2k$-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into $k$ edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every $2k$-valent connected circulant graph has a decomposition into $k$ edge-disjoint Hamilton cycles. We settle the problem of decomposing $2k$-valent infinite circulant graphs into $k$ edge-disjoint two-way-infinite Hamilton paths for $k=2$, in many cases when $k=3$, and in many other cases including where the connection set is $\pm\{1,2,\ldots,k\}$ or $\pm\{1,2,\ldots,k-1,k+1\}$.

connected Cayley graph has a Hamilton cycle, see [26]. Both the above-mentioned conjectures remain open.
A decomposition of a graph is a set of edge-disjoint subgraphs that collectively contain all the edges; a decomposition into Hamilton cycles is called a Hamilton decomposition, and a graph admitting a Hamilton decomposition is said to be Hamilton-decomposable. An obvious necessary condition for a Hamilton decomposition of a graph is that the graph be regular of even valency. Sometimes, a decomposition of a (2 + 1)-valent graph into Hamilton cycles and a perfect matching is also called a Hamilton decomposition, but here we do not consider these to be Hamilton decompositions.
In 1984, Alspach [1] asked whether every 2 -valent connected Cayley graph on a finite abelian group is Hamilton-decomposable. It is known that every connected Cayley graph on a finite abelian group has a Hamilton cycle [7], so it makes sense to consider the stronger property of Hamilton-decomposability. Alspach's question is now commonly referred to as Alspach's conjecture. It holds trivially when = 1 and Bermond et al. proved that it holds for = 2 [3]. The case = 3 is still open, although many partial results exist, see [8,9,[22][23][24]. There are also results for > 3, see [2,12,[17][18][19]. It was shown in [4] that there exist 2 -valent connected Cayley graphs on finite non-abelian groups that are not Hamilton-decomposable.
In this article, we study the natural extension of Alspach's question to the case of Cayley graphs on infinite abelian groups, specifically in the case of the infinite cyclic group ℤ. We will not be considering any uncountably infinite graphs, so it should be assumed that the order of any graph in this article is countable. The natural infinite analog of a (finite) Hamilton cycle is a two-way-infinite Hamilton path, which is defined as a connected spanning 2-valent subgraph. This is, of course, an exact definition for a Hamilton cycle in the finite case, and accordingly we define a Hamilton decomposition of an infinite graph to be a decomposition into two-way-infinite Hamilton paths. A one-way-infinite Hamilton path is a connected spanning subgraph in which there is exactly one vertex of valency 1, and the remaining vertices have valency 2. For convenience, since we will not be dealing with one-way-infinite Hamilton paths, we refer to two-way-infinite Hamilton paths simply as Hamilton paths, or as infinite Hamilton paths if we wish to emphasize that the path is infinite.
Hamiltonicity of infinite circulant graphs, and of infinite graphs generally, has already been studied. In 1959, Nash-Williams [21] showed that every connected Cayley graph on a finitely generated infinite abelian group has a Hamilton path. It seems that Nash-Williams' article is largely unknown. For example, it is not cited in the 1984 survey by Witte and Gallian [26], and in 1995 Zhang and Huang [27] proved the above-mentioned result of Nash-Williams in the special case of infinite circulant graphs. Indeed, D. Jungreis' article [15] on Hamilton paths in infinite Cayley digraphs is one of the few articles to cite Nash-Williams' result. Other results on Hamilton paths in infinite Cayley digraphs can be found in [11,16].
Given the existence of Hamilton paths in Cayley graphs on finitely-generated infinite abelian groups, it makes sense to consider Hamilton-decomposability of these graphs. In this article, we investigate this problem in the special case of infinite circulant graphs. Witte [25] proved that an infinite graph with infinite valency has a Hamilton decomposition if and only if it has infinite edge-connectivity and has a Hamilton path. By combining this characterization with the result of Nash-Williams, we observe that if a connected Cayley graph on a finitely generated infinite abelian group has infinite valency, then it is Hamilton-decomposable, see Theorem 8.
In Lemma 2, we prove necessary conditions for an infinite circulant graph to be Hamiltondecomposable, thereby showing that not all connected infinite circulant graphs are Hamiltondecomposable. Since there are no elements of order 2 in ℤ, any infinite circulant graph with finite connection set is regular of valency 2 and is 2 -edge-connected, for some nonnegative integer . Thus, neither the valency nor the edge-connectivity is an immediate obstacle to Hamilton-decomposability.
We call infinite circulant graphs admissible if they satisfy the necessary conditions for Hamiltondecomposability given in Lemma 2 (see Definition 3). In Section 3, we prove that all admissible 4-valent infinite circulant graphs are Hamilton-decomposable. We also show, in Section 4, that several other infinite families of infinite circulant graphs are Hamilton-decomposable, including many 6-valent infinite circulant graphs, and several families with arbitrarily large finite valency.
Throughout the article we make use of the following notation and terminology. Let  be a group with identity and ⊆  − { } that is inverse-closed, that is, −1 ∈ if and only if ∈ . The Cayley graph on the group  with connection set , denoted Cay(, ), is the undirected simple graph whose vertices are the elements of  and whose edge set is {{ , } | ∈ , ∈ }. When  is an infinite group, we call Cay(, ) an infinite Cayley graph. When  is a cyclic group, a Cayley graph Cay(, ) is called a circulant graph. Since we are interested in infinite circulant graphs, we will be considering graphs Cay(ℤ, ), where is an inverse-closed set of distinct nonzero integers, which may be finite or infinite. We define If is any subset of ℤ and ∈ ℤ, then we write + to represent the set { + | ∈ }. Furthermore, if is any graph with ( ) ⊆ ℤ and ∈ ℤ, then + is the graph with vertex set { + | ∈ ( )} and edge set {{ + , + } | { , } ∈ ( )}. The length of any edge { , }, denoted ( , ), in a graph with vertex set ℤ or ℤ is the distance from to in Cay(ℤ, {±1}) or Cay(ℤ , {±1}) if the vertex set is ℤ or ℤ , respectively. Next we discuss some notation for walks and paths in infinite circulant graphs that we will use throughout the remainder of the article. The finite path with vertex set { 1 , 2 , … , } and edge set For ∈ ℤ and 1 , 2 , … , ∈ , we define Ω ( 1 , 2 , … , ) to be the walk in Cay(ℤ, ) where the sequence of vertices is , , so the lengths of the edges in the walk are | 1 |, | 2 |, … , | |. Whenever we write Ω ( 1 , 2 , … , ) it will be the case that [ , + 1 , … , + ∑ =1 ] is a path, and we will use the notation Ω ( 1 , 2 , … , ) interchangeably for both the walk (with associated orientation, start, and end vertices) and the path [ , + 1 , … , + ∑ =1 ] (which is a graph with no inherent orientation).

NECESSARY CONDITIONS AND INFINITE VALENCY
The following two lemmas give a characterization of connected infinite circulant graphs, and necessary conditions for an infinite circulant graph to be Hamilton-decomposable. We remark that the main idea in the proof of Lemma 2 has been used in [5,13].

Lemma 2. If Cay(ℤ, ) is Hamilton-decomposable, then
Proof. Suppose Cay(ℤ, ) has Hamilton decomposition . A Hamilton-decomposable graph is clearly either empty or connected, and so (i) follows immediately from Lemma 1. If is finite, then let = | + | and let = {{ , } ∈ (Cay(ℤ, )) | ≤ 0, ≥ 1}. For each ∈ + , there are exactly edges of length in , and so we have | | = ∑ ∈ + . However, it is clear that each of the Hamilton paths in  has an odd number of edges from . This means that | | ≡ (mod 2), and (ii) holds. ■ It may seem plausible to use a Hamilton decomposition of a finite circulant graph to construct a Hamilton decomposition of an infinite circulant graph whose edges have the same lengths as those of the finite graph. However, this is not possible in general.
For example, Cay(ℤ , ±{1, 2}) is Hamilton-decomposable for every ≥ 5, yet Cay(ℤ, ±{1, 2}) is not Hamilton-decomposable by Lemma 2. There are infinitely many connected infinite circulant graphs that are not admissible (and thus not Hamilton-decomposable). For example, if is finite and Cay(ℤ, ) is admissible, then for every even positive integer ∉ + , Cay(ℤ, ∪ ±{ }) is not admissible. We have found no admissible infinite circulant graphs that are not Hamilton-decomposable, and thus we pose the following problem.

Problem 4. Is every admissible infinite circulant graph Hamilton-decomposable?
We now show that results from [25] and [21] combine to settle this problem for the case where is infinite. An infinite graph with infinite valency is ∞-connected if ⧵ is connected for every finite subset ⊂ ( ) (that is, has no finite cut-set). An infinite graph with infinite valency has infinite edge-connectivity if ⧵ is connected for every finite subset ⊂ ( ) (that is, has no finite edge-cut). It is easy to see that if is ∞-connected then has infinite edge-connectivity, but the converse of this does not hold.

Theorem 8. A Cayley graph of infinite valency on a finitely-generated infinite abelian group is Hamilton-decomposable if and only if it is connected.
Proof. Let be a Cayley graph of infinite valency on a finitely generated infinite abelian group. If is Hamilton-decomposable, then clearly it is connected. For the converse, suppose is connected. By Theorem 7, has a Hamilton path. Since is a Cayley graph, it is vertex-transitive. Thus, is ∞-connected by Theorem 5, and hence has infinite edge-connectivity. So is Hamilton-decomposable by Theorem 6. ■ Since an infinite circulant graph with infinite connection set is admissible if and only if it is connected, Theorem 8 answers Problem 4 in the affirmative for the case of infinite connection sets. For the remainder of this article, we consider the case where the connection set is finite.

INFINITE 4-VALENT CIRCULANT GRAPHS
In this section, we prove that all admissible 4-valent infinite circulant graphs are Hamiltondecomposable, thus establishing the following theorem.
For ) ∪ , see Figure 1. It is straightforward to check that is a path with endpoints 0 and 2 having 2 edges. Since gcd( , ) = 1 and the lengths of the edges of alternate between and , it follows that has exactly one vertex from each congruence class modulo 2 (except that the endpoints are both 0 (mod 2 )). Hence, 1 = ⋃ ∈ℤ ( + 2 ) is a Hamilton path in . The second Hamilton path is 2 = ⋃ ∈ℤ ( + 2 ), where = + . To see that 1 and 2 are edge-disjoint, it suffices to show that ( ) is disjoint from both ( ) and ( + 2 ). Observe that the edge set of is:

INFINITE -VALENT CIRCULANT GRAPHS
In this section, we consider 2 -valent infinite circulant graphs for ≥ 3, proving that there are many infinite families of such graphs that are Hamilton decomposable if and only if they are admissible. We begin by considering graphs Cay(ℤ, ) where | + | = and ∈ + and no other element of + is divisible by . An example of such a graph is Cay (ℤ, ±{1, 5, 11, 12, 14}). The next lemma, Lemma 10, shows that in order to find a Hamilton decomposition of an admissible infinite circulant graph of this type, it suffices to find a Hamilton path in the complete graph with vertex set ℤ with edges of appropriate lengths. In the following lemma and its proof, the notation [ ] denotes the congruence class of modulo . Let Σ = ∑ =1 and note that 1 is a path in with endpoints 0 and Σ, and that 1 has exactly one vertex from each congruence class modulo . Similarly, 1 + is a path in with endpoints and + Σ, 1 + has exactly one vertex from each congruence class modulo , and 1 + is disjoint from 1 . Now define to be the path in with edge set Thus, is a path with endpoints 0 and 2 having 2 edges and having exactly one vertex from each congruence class modulo 2 (except that the the endpoints are both 0 (mod 2 )). Thus, is a Hamilton path in .
Thus, is a path with endpoints 0 and having exactly one vertex from each congruence class modulo (except that the endpoints are both 0 (mod )). Hence 1 = ⋃ ∈ℤ ( + ) is a Hamilton path in . For each ∈ {1, 2, … , − 1, + 1} the path uses exactly one edge of the form { , + }. Thus { 1 , 1 + 1, 1 + 2, … , 1 + − 1} is a Hamilton decomposition of . ■ Next, we consider graphs Cay(ℤ, ) where + consists of consecutive even integers together with 1. In either case, is a path with endpoints 0 and having exactly one vertex from each congruence class modulo (except that the endpoints are both 0 (mod )). We divide the proof into three cases, odd, ≡ 0 (mod 4), and ≡ 2 (mod 4) and provide a construction for a Hamilton decomposition in each case. The constructions are based on a starter path that has 3 edges, using edges of each length 1, 2, and 2 . In each case it can be checked that is a Hamilton path and that { 1 , 1 + , 1 + 2 } is a Hamilton decomposition of .
In the cases below, we use the following notation: