Factorizing the Rado graph and infinite complete graphs

Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of infinite graphs, together with $\Lambda$. The Factorization Problem $FP(\mathcal{F}, \Lambda)$ asks whether $\mathcal{F}$ can be realized as a factorization of $\Lambda$, namely, whether there is a factorization $\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of $\Lambda$ such that each $\Gamma_{\alpha}$ is a copy of $F_{\alpha}$. We study this problem when $\Lambda$ is either the Rado graph $R$ or the complete graph $K_\aleph$ of infinite order $\aleph$. When $\mathcal{F}$ is a countable family, we show that $FP(\mathcal{F}, R)$ is solvable if and only if each graph in $\mathcal{F}$ has no finite dominating set. We also prove that $FP(\mathcal{F}, K_\aleph)$ admits a solution whenever the cardinality $\mathcal{F}$ coincide with the order and the domination numbers of its graphs. For countable complete graphs, we show some non existence results when the domination numbers of the graphs in $\mathcal{F}$ are finite. More precisely, we show that there is no factorization of $K_{\mathbb{N}}$ into copies of a $k$-star (that is, the vertex disjoint union of $k$ countable stars) when $k=1,2$, whereas it exists when $k\geq 4$, leaving the problem open for $k=3$. Finally, we determine sufficient conditions for the graphs of a decomposition to be arranged into resolution classes.


Introduction
We assume that the reader is familiar with the basic concepts in (infinite) graph theory, and refer to [10] for further details.
In this paper all graphs will be simple, namely, without multiple edges or loops.As usual, we denote by V (Λ) and E(Λ) the vertex set and the edge set of a simple graph Λ, respectively.We say that Λ is finite (resp.infinite) if its vertex set is so, and refer to the cardinality of V (Λ) and E(Λ) as the order and the size of Λ, respectively.Note that in the finite case |E(Λ)| ≤ |V (Λ)| 2 , whereas if Λ is infinite, then its order, which is a cardinal number, is greater than or equal to its size.We use the notation K v for any complete graph of order v, and denote by K V the complete graph whose vertex set is V .
Given a subgraph Γ of a simple graph Λ, we denote by Λ \ Γ the graph obtained from Λ by deleting the edges of Γ.If Γ contains all possible edges of Λ joining any two of its vertices, then Γ is called an induced subgraph of Λ (in other words, an induced subgraph is obtained by vertex deletions only).Instead, if V (Γ) = V (Λ), then Γ is called a spanning subgraph or a factor of Λ (hence, a factor is obtained by edge deletions only).If Γ is also h-regular, then we speak of an h-factor.We recall that a set D of vertices of Λ is dominating if all other vertices of Λ are adjacent to some vertex of D. The minimum size of a dominating set of Λ is called the domination number of Λ.Finally, we say that Λ is locally finite if its vertex degrees are all finite.
A decomposition of Λ is a set G = {Γ 1 , . . ., Γ n } of subgraphs of Λ whose edgesets partition E(Λ).If the graphs Γ i are all isomorphic to a given subgraph Γ of Λ, then we speak of a Γ-decomposition of Λ.When Γ and Λ are both complete graphs, we obtain 2-designs.More precisely, a K k -decomposition of K v is equivalent to a 2-(v, k, 1) design.
Classically, the graphs Γ i and Λ are taken to be finite, and the same usually holds for the parameters v and k of a 2-designs.However, there has been considerable interest in designs on a infinite set of v points, mainly when k = 3.In this case, we obtain infinite Steiner triple systems whose first explicit constructions were given in [12,13].Further results concerning the existence of rigid, sparse, and perfect countably Steiner triple systems can be found in [6,7,11].Results showing that any Steiner system can be extended are given in [1,15].The existence of large sets of Steiner triple systems for every infinite v (and more generally, of infinite Steiner systems) can be found in [4].Also, infinite versions of topics in finite geometry, including infinite Steiner triple systems and infinite perfect codes are considered in [3].A more comprehensive list of results on infinite designs can be found in [9].
When each graph of a decomposition G of Λ is a factor (resp. h-factor), we speak of a factorization (resp.h-factorization) of Λ.Also, when the factors of G are all isomorphic to Γ, we speak of a Γ-factorization of Λ.A factorization of K v into factors whose components are copies of K k is equivalent to a resolvable 2-(v, k, 1) design.
In this paper, we consider the Factorization Problem for infinite graphs, which is here stated in its most general version Problem 1.1.Let Λ be a graph of order ℵ and let F = {F α : α ∈ A} be a family of (non-empty) infinite graphs, not necessarily distinct, each of which has order ℵ, with ℵ ≥ |A|.
The Factorization Problem F P (F , Λ) asks for a factorization G = {Γ α : α ∈ A} of Λ such that Γ α is isomorphic to F α , for every α ∈ A. If Λ is the complete graph of order ℵ, we simply write F P (F ).If in addition to this each F α is isomorphic to a given graph F and |A| = ℵ, we write F P (F ). 1s far as we know, there are only four papers dealing with the Factorization Problem for infinite complete graphs, and two of them, concern classic designs.In [14] it is shown that there exists a resolvable 2-design whenever v = |N| and k is finite; these designs have, in addition, a cyclic automorphism group G acting sharply transitive on the vertex set; briefly they are G-regular.In [9] it is shown that every infinite 2-design with k < v is necessarily resolvable, and when k = v, both resolvable and non-resolvable designs exist.We point out that both these papers deal with t-designs with t ≥ 2.
Furthermore, in [2] the authors construct a G-regular 1-factorization of a countable complete graph for every finitely generated abelian infinite group G. Finally, [8] proves the following.
Theorem 1.2.Let F be a graph whose order is the cardinal number ℵ. F P (F ) has a G-regular solution whenever the following two conditions hold: G is an involution free group of order ℵ.
Note that this result generalizes the one obtained in [14] to any complete graph of infinite order ℵ, blocks of any size less than ℵ, and groups G not necessarily cyclic.Furthermore, Theorem 1.2 can also be seen as a generalization of the result in [2] to complete graphs of any infinite order.
When dealing with infinite graphs, a central role is played by the Rado graph R (see [16]), named after Richard Rado who gave one of its first explicit constructions.Indeed, R is the unique countably infinite random graph, and it can be constructed as follows: V (R) = N and a pair {i, j} with i < j is an edge of R if and only if the i-th bit of the binary representation of j is one.R shows many interesting properties, such as the universal property: every finite or countable graph can be embedded as an induced subgraph of R.
When replacing the concept of induced subgraph with the dual one of factor, a weaker result holds.Indeed, in [5] it is pointed out that a countable graph F can be embedded as a factor of R if and only if the domination number of F is infinite.In the same paper, it is further shown that F P (F , R) has a solution whenever F is infinite and each of its graphs is locally finite.Note that a locally finite countable graph has infinite domination number, but the converse is not true: for example, the Rado graph is not locally finite and it has no finite dominating set (indeed, for every D = {i 1 , . . ., i t } ⊂ N, there exists an integer j ∈ N whose binary representation has 0 in positions i 1 , . . ., i t , which means that j is adjacent with no vertex of D).
In this paper, we extend this result to any countable family F of admissible graphs.More precisely, we prove the following.
Theorem 1.3.Let F be a countable family of countable graphs.Then, F P (F , R) has a solution if and only if the domination number of each graph of F is infinite.
Furthermore, we prove the solvability of F P (F ) whenever the size of F coincides with the order and the domination number of its graphs.
Theorem 1.4.Let F be a family of graphs, each of which has order ℵ.F P (F ) has a solution whenever the following two conditions hold: 2. the domination number of each graph in F is ℵ.
When F contains only copies of a given graph F satisfying condition 1 of Theorem 1.2 (i.e., F is locally finite), then F satisfies both conditions 1 and 2 of Theorem 1.4.Therefore, Theorem 1.4 can be seen as a generalization of Theorem 1.2, even though it does not provide any information on the automorphisms of a solution to FP.
Note that if we just require that the domination number of each graph of F is ℵ, there may exist factorizations with fewer factors than ℵ; this means that the two conditions in Theorem 1.4 are independent.Indeed, the Rado graph R has no finite dominating set and Corollary 2.4 shows that for every n ≥ 2 there exists a factorization of K N into n copies of R. We point out that Theorem 1.4 constructs instead factorizations of K N into infinite copies of R.
The paper is organized as follows.In Sections 2 and 3, we prove the main results of this paper, Theorems 1.3 and 1.4.In Section 4, we deal with Ffactorizations of K N when F belongs to a special class of graphs with finite domination number (and hence not satisfying condition 2 of Theorem 1.4): the countable k-stars (briefly, S k ), that is, the vertex disjoint union of k countable stars.We prove that F P (S k ) has a solution whenever k > 3, and there is no solution for k ∈ {1, 2}.This shows that there are families F of graphs for which F P (F ) is not solvable.We leave open the problem when k = 3.In the last section, inspired by [9], we provide a sufficient condition for a decomposition F of K ℵ to be resolvable (i.e., the graphs of F can be partitioned into factors of K ℵ ).

Factorizing the Rado graph
In this section, we prove Theorem 1.3.Also, since the Rado graph R is selfcomplementary, that is, K N \ R is isomorphic to R, we obtain as a corollary the countable version of Theorem 1.4.
We start by recalling an important characterization of the Rado graph (see, for example, [5]).
Theorem 2.1.A countable graph is isomorphic to the Rado graph if and only if it satisfies the following property: ⋆ for every disjoint finite sets of vertices U and W , there exists a vertex z adjacent to all the vertices of U and non-adjacent to all the vertices of V .
Now we slightly generalize the construction of the Rado graph given in the introduction.
Definition 2.2.Given a set I ⊂ {0, . . ., q − 1}, with 1 ≤ |I| < q, we denote by R q I the following graph: V (R q I ) = N, and {x, y}, with x < y, is an edge of R q I whenever the x-th digit of y in the base q expansion of y belongs to I.
Proposition 2.3.Every graph R q I is isomorphic to the Rado graph.Proof.By Theorem 2.1, it is enough to show that property ⋆ holds for R q I .We assume, without loss of generality, that 0 ∈ I while 1 ∈ I, and let U and V be two disjoint subsets of N. Then there are infinitely many positive integers whose base q expansion has 0 in each position u ∈ U and 1 in each position v ∈ V .Denoting by z one of these integers larger than max(U ∪ V ), we have that z is adjacent to all the vertices of U but to none in V .
Note that K N = q−1 i=0 R q {i} and R q {0,...,q−2} = q−2 i=0 R q i .Considering that the R q {i} s are pairwise edge-disjoint and isomorphic to the Rado graph, we obtain the following.
Corollary 2.4.For every n ∈ N, the graphs R and K N can be factorized into n and n + 1 copies of R, respectively.
The following result is crucial to prove Theorem 1.3.It strengthens a result given in [5] and allows us to suitably embed in the Rado graph R any countable graph with infinite domination number.
Proposition 2.5.Let F be a countable graph with no finite dominating set.For every edge e ∈ E(R), there exists an embedding σ e of F in R such that: We can assume without loss of generality that e lies in R 3 {0} .In [5,Proposition 8], it is shown that there exists an embedding σ e of F into the Rado graph R 3 {0} ⊂ R satisfying condition 1.It is then left to prove that condition 2 holds.By Theorem 2.1, this is equivalent to saying that R \ σ e (F ) satisfies ⋆.
Let U and V be two finite disjoint subsets of N. Clearly, there are infinitely many positive integers whose base 3 expansion has 1 in each position u ∈ U and 2 in each position v ∈ V .Let z be one of these integers larger than max(U ∪ V ).Since R \ σ e (F ) contains R 3 {1} and it is edge-disjoint with R 3 {2} , it follows that z is adjacent in R \ σ e (F ) to all the vertices of U and is non-adjacent to all the vertices of V .This means that ⋆ holds for R \ σ e (F ).
We are now ready to prove Theorem 1.3, whose statement is recalled here, for clarity.
Theorem 1.3.Let F be a countable family of countable graphs.Then, F P (F , R) has a solution if and only if the domination number of each graph of F is infinite.
Proof.Since the Rado graph has no finite dominating set, the same holds for its spanning subgraphs.Hence, each graph of F must have infinite domination number.Under this assumption, we are going to show that F P (F , R) has a solution.
Let E(R) = {e 1 , . . ., e n , . ..} and F = {F 1 , . . ., F n , . ..}.By recursively applying Proposition 2.5, we obtain a sequence of isomorphisms σ i : F i → Γ i satisfying for each i ∈ N the following properties: It follows that the Γ i s are pairwise edge-disjoint factors of R which partition E(R).Therefore, {Γ i : i ∈ N} is a solution to F P (F , R).
The proof of Theorem 1.3 allows us to construct solutions to F P (F , R) even when the cardinality of F is finite, provided that F contains a copy of the Rado graph.In other words, we have the following.
Corollary 2.6.Let F be a finite family of countable graphs such that 1. F contains at least one graph isomorphic to the Rado graph; 2. the domination number of each graph in F is infinite.
Then, F P (F , R) has a solution.
Recalling that R is self complementary, the countable version of Theorem 1.4 can be easily obtained as a corollary to Theorem 1.3.
Corollary 2.7.Let F be a countable family of countable graphs.F P (F ) has a solution whenever the domination number of each graph in F is infinite.
Proof.Recall that R 2 {0} and R 2 {1} are copies of R which together factorize K N .Therefore, it is enough to partition F into two countable families F 1 and F 2 , and then apply Theorem 1.3 to get a solution The natural generalization of property ⋆ to a generic cardinality ℵ is the following one: ⋆ ℵ for every disjoint sets of vertices U and W whose cardinality is smaller than ℵ, there exists a vertex z adjacent to all the vertices of U and nonadjacent to all the vertices of V .
Then, using the transfinite induction (see Theorem 3.5 below), one could also prove the following generalization of Proposition 2.1: Proposition 2.8.Any two graphs of order ℵ that satisfy property ⋆ ℵ are pairwise isomorphic.
Therefore, we can refer to any graph of order ℵ and satisfying property ⋆ ℵ as the ℵ-Rado graph R ℵ .Its existence is guaranteed under the Generalized Continuum Hypothesis (GCH) which states that if ℵ ′ ≺ ℵ then 2 ℵ ′ ℵ.Under GCH, one can see that the set of all q-ary sequences of length ≺ ℵ has size ℵ: indeed, for every ℵ ′ ≺ ℵ, the set of all q-ary sequences of length ℵ ′ has cardinality 2 ℵ ′ , and by GCH we have that 2 ℵ ′ ℵ.This means that the construction of the countable Rado graph (Definition 2.2) based on representing every natural number with a finite q-ary sequence (its base q expansion) can be generalized to any order.
By assuming that GCH holds, we can prove the following generalization of Theorem 1.3.Theorem 2.9.Let F be a family of graphs of order ℵ and assume that |F | = ℵ.Then F P (F , R ℵ ) has a solution if and only if the domination number of each graph in F is ℵ.

Factorizing infinite complete graphs
In this section we prove Theorem 1.4.We point out that if we assume the Generalized Continuum Hypothesis, considering that R ℵ is self complementary by property ⋆ ℵ , we can proceed as in Corollary 2.7 and obtain Theorem 1.4 as a consequence of Theorem 2.9.
Here we present a proof of Theorem 1.4 that does not require GCH, which we recall is independent of ZFC.Therefore, a proof that does not require GCH is to be preferred.
We say that a graph or a set of vertices is ℵ-small (resp.ℵ-bounded) if their order or cardinality is smaller than ℵ (resp.smaller than or equal to ℵ).Given two graphs F and Λ of order ℵ, we denote by Σ ℵ (F, Λ) the set of all graph embeddings between an induced ℵ-small subgraph of F and a subgraph of Λ.A partial order on Σ ℵ (F, Λ) can be easily defined as follows: if σ : G → Γ and σ ′ : G ′ → Γ ′ are embeddings of Σ ℵ (F, Λ), we say that σ ≤ σ ′ whenever σ ′ is an extension of σ, namely, G and Γ are subgraphs of G ′ and Γ ′ , respectively, and Lemma 3.1.Let F be a graph of order ℵ and with no ℵ-small dominating set.Also, let Θ be an ℵ-small subgraph of K ℵ , and let σ ∈ Σ ℵ (F, K ℵ \ Θ).
1. Let G ′ be the subgraph of F induced by v and V (G).Since 2. Since F has no ℵ-small dominating set, V (G) (which is an ℵ-small set) cannot be a dominating set for F .Hence, there is a vertex a ∈ V (F ) that is not adjacent to any of the vertices of G.We denote by G ′′ (resp., Γ ′′ ) the graph obtained by adding a to G (resp., x to Γ) as an isolated vertex.
Clearly, G ′′ is an induced subgraph of F ; also, Γ ′′ and Θ have no edge in common, since E(Γ ′′ ) = E(Γ).Therefore, the extension σ ′′ : From now on, we will work within the Zermelo-Frankel axiomatic system with the Axiom of Choice in the form of the Well-Ordering Theorem.We recall the definition of a well-order.Definition 3.2.A well-order ≺ on a set X is a total order on X with the property that every non-empty subset of X has a least element.
The following theorem is equivalent to the Axiom of Choice.Theorem 3.3 (Well-Ordering).Every set X admits a well-order ≺.
Given an element x ∈ X, we define the section X ≺x associated to it: Corollary 3.4.Every set X admits a well-order ≺ such that the cardinality of any section is smaller than |X|.
Proof.Let us consider a well-order ≺ on X.Let x be the smallest element such that X ≺x has the same cardinality as X.The set Y = X ≺x is such that all its sections with respect to the order ≺ have smaller cardinality.Since Y instead has the same cardinality as X, the order ≺ on Y induces an order ≺ ′ on X with the required property.
We recall now that well-orderings allow proofs by induction.Theorem 3.5 (Transfinite induction).Let X be a set with a well-order ≺ and let P x denote a property for each x ∈ X. Set 0 = min X and assume that: • P 0 is true, and • for every x ∈ X, if P y holds for every y ∈ X ≺x , then P x holds.
Then P x is true for every x ∈ X.
We are now ready to prove Theorem 1.4.The idea behind the proof can be better understood by restricting our attention to the countable case, ℵ = N.To solve F P ({F α : α ∈ N}), we first order the edges of K N : {e 0 , e 1 , . . ., e γ , . ..}.Then, we define embeddings σ β α : G β α → Γ β α where G β α is an induced subgraph of F α , and Γ β α is a subgraph of K N .These embeddings are obtained by recursively applying Lemma 3.1 which adds, at each step, a vertex to G β α and a vertex to Γ β α and makes sure that the vertex β belongs to both these graphs (this procedure can be seen as a variation of Cantor's "back-and-forth" method).We also make sure that, for every γ, the graphs Γ γ 0 , Γ γ 1 , . . ., Γ γ γ are pairwise edge-disjoint and contain between them the edge e γ .The solution to F P ({F α : α ∈ N}) will be represented by G = {Γ α : α ∈ N} where Γ α = β Γ β α .
Theorem 1.4.Let F be a family of graphs, each of which has order ℵ.F P (F ) has a solution whenever the following two conditions hold: . the domination number of each graph in F is ℵ.
To prove the assertion, we construct a chain of families (E γ ) γ∈A , where γ, and the following three conditions: γ ) for every β ∈ A γ , the graphs Γ β α : α β are pairwise edge-disjoint, and the edge e β belongs to their union; (3 γ ) for every α, β ∈ A γ , the graph Γ β α is either finite or |A γ |-bounded.The desired factorization of K ℵ is then G = {Γ α : α ∈ A}, where Γ α = β∈A Γ β α for every α ∈ A. Indeed, properties (1 γ ) guarantee that each Γ α is a factor of K ℵ isomorphic to F α Also, properties (2 γ ) ensure that the Γ α s are pairwise edge-disjoint and between them contain all the edges of K ℵ .
• Base case.If e γ ∈ Θ, let σ be the empty map of Σ ℵ (F γ , K ℵ \ Θ).Otherwise, chose an edge e ∈ E(F γ ), and let σ ∈ Σ ℵ (F γ , K ℵ \ Θ) be the embedding that maps e to e γ .By Lemma 3.1, there exists σ 0 γ : • Inductive step.Assume we have defined the maps σ β ′ γ for any β ′ ≺ β.Again by Lemma 3.1, there exists σ β γ : It follows from the construction that the family 4 The Factorization Problem for k-stars Theorem 1.4 does not provide solutions to F P (F ) whenever the graph F has a dominating set of cardinality less than its order.In particular, if F is countable with a finite dominating set, then the existence of a solution to F P (F ) is an open problem.In this section, we consider a special class of such graphs, the k-stars S k .More precisely, • the star S 1 is the graph with vertex-set N whose edges are of the form {0, i} for every i ∈ N \ {0}; • the k-star S k is the vertex-disjoint union of k stars.
Note that S k contains exactly k vertices of infinite degree, which we call centers and form a finite dominating set of S k .
In the following, we show that F P (S k ) has no solution whenever k ∈ {1, 2}, while it admits a solution for every k > 3. Unfortunately, we leave open the problem for 3-stars.Proof.Assume for a contradiction that there is a factorization G of K N into 1stars.Choose any star Γ ∈ G and let g denote its center.Considering that all the edges of K N incident with g belong Γ, it follows that g cannot be a vertex in any other star of G, which therefore are not factors and this is a contradiction.

The case k ∈ {1, 2}
With essentially the same proof, one obtains the following.
We show that Γ ′ 1,h is an h-star satisfying condition (1) by induction on h.
1 is a 2-star with the same vertex-set as Γ 1 .Now assume that Γ ′ 1,h is an h-star satisfying condition (1) for some h ≥ 2. Recalling the definition of Γ ′ 1,h and Γ ′ 1,h+1 , and considering that the vertex-sets of ∆ h−1 and ∆ * h partition V (∆ * h−1 ), we have that Γ ′ 1,h+1 is an (h + 1)-star with the same vertex-set as Γ ′ 1,h , that is, V (Γ 1 ), and this concludes the proof.Propositions 4.1, 4.4 and 4.5 leave open F P (S k ) only when k = 3.In this case, an approach similar to Theorem 4.5 cannot work, as shown in the following.Proposition 4.6.There is no 3-star Γ with vertex-set V = Z × {0, 1, . . ., k} such that the Z-orbit of Γ is an S 3 -factorization of K V Proof.Assume for a contradiction that there exists a 3-star Γ with vertex-set We first notice that Γ must have at least a center in Z × {i}, for every i ∈ {0, 1, . . ., k}.Indeed, if Γ has no center in Z × {i} for some i ∈ {0, 1, . . ., k}, then no edge of K Z×{i} can be covered by G. Since Γ has 3 centers, it follows that k ≤ 2. Note that if k = 2, the centers of Γ must be x 0 , y 1 , z 2 for some x, y, z ∈ Z, but in this case the edge {x 0 , y 1 } cannot lie in any translate of Γ. Therefore k ≤ 1.
If k = 1, without loss of generality we can assume that the centers of Γ are 0 0 , x 1 and y 1 with x = y.Since the edge {0 0 , x 1 } does not belong to Γ, it lies in some of its translates, say Γ + z with z = 0.This is equivalent to saying that Similarly, we can show that {(x − y) 0 , x 1 } ∈ Γ.It follows that Γ cannot contain the edges {0 0 , (x− y) 0 } and {0 0 , (y − x) 0 }.This implies that no edge of the form {w 0 , (x − y + w) 0 } lie in any translate of Γ, contradicting again the assumption that G is a factorization of K V .Therefore k = 0.

The resolvability problem
Furthermore, if the domination number of some graph Proof.For every α ∈ A, set ℵ α = ℵ if the domination number of F α is less than ℵ; otherwise, let 0 ℵ α ℵ.By adding to each graph F α a set of ℵ α isolated vertices we obtain a graph F ′ α whose order and domination number are ℵ.Since the assumptions of Theorem 1.4 are satisfied, there exists a factorization G ′ = {Γ ′ α : α ∈ A} of K ℵ such that each Γ ′ α is isomorphic to F ′ α .By replacing Γ ′ α with the isomorphic copy of F α , we obtain the desired decomposition G.
Inspired by [9], we ask under which conditions a decomposition G of K ℵ is resolvable, namely, its graphs can be partitioned into factors of K ℵ , also called resolution classes.It follows that a resolvable decomposition G of K ℵ must satisfy the following two conditions: In the following, we easily construct decompositions of K ℵ that do not satisfy the above conditions, and therefore they are non-resolvable.
For instance, if ℵ = |N|, each F α is a countable locally finite graph (hence, its domination number is ℵ) and ℵ ′ = 1 for every β ∈ N, then we construct a decomposition G = {G β : β ∈ N} of K N into connected regular graphs where V (G β ) = N \ {x β } for some x β ∈ N. Clearly, no graph of G is a factor of K N , and any two graphs of G have common vertices.Therefore, G is not resolvable.
Example 5.3.Let G be any decomposition of the infinite complete graph K V (for example, one of those constructed by Corollary 5.1).Let y and z be vertices not belonging to K V and set W = V ∪ {y, z}.We can easily extend G to a non-resolvable decomposition G ′ of K W in the following way.
Choose x ∈ V and let C be the following family of paths of length 1 or 2: Also, x, y and z do not satisfy condition N2, since Therefore, G ′ is nonresolvable.Indeed, any resolution class of G ′ could cover the vertex z only with graphs passing through x or y.This means that the graph [x, y] cannot belong to any resolution class of G ′ .
The following result provides sufficient conditions for a decomposition G to be resolvable.Theorem 5.4.Let G be a decomposition of the infinite complete graph K ℵ satisfying the following properties for some ℵ ′ ≺ ℵ: Then G is resolvable.
Proof.Let G = {G α : α ∈ A}.We consider a well-order ≺ on A satisfying Corollary 3.4.Since the graphs of G are ℵ ′ -bounded, we have that |A| = ℵ and we can assume V (K ℵ ) = A.Here we need to construct an ascending chain (G γ ) γ∈A of families G γ := {Γ γ α : α ∈ A γ } (where Γ γ ′ α is a subgraph of Γ γ α whenever γ ′ γ) that satisfy the following proprieties: α for every α ∈ A. Indeed, due to properties (1 γ ) and (2 γ ), each Γ α is a resolution class of G and, by property (3 γ ), R is a partition of G into resolution classes.
We proceed by transfinite induction on γ.
TRANSFINITE INDUCTIVE STEP.For every γ ′ ≺ γ, we assume there is a family G γ ′ satisfying (i γ ′ ) for 1 ≤ i ≤ 4. We show that G γ ′ can be extended to a family G γ that satisfies the same properties, (i γ ) for 1 ≤ i ≤ 4.
• Base case.Let us first suppose that G γ is not contained in any Γ γ α ′ (where α ′ ≺ γ).Again, by conditions R1 and R2, there exists G ∈ G(0) that is also not contained in any Γ γ α ′ such that G is either G γ or is disjoint from G γ .We set Γ 0 γ to be G γ ∪ G. Otherwise, we set Γ 0 γ to be any graph G in G(0) that is not contained in any Γ γ α ′ .
Remark 5.5.A cardinal ℵ is said to be regular if any ℵ-small union of ℵ-small sets (resp.graphs) is still an ℵ-small set (resp.graph) otherwise it is said to be singular.It is easy to see that, for regular cardinals, conditions R1 and R2 of Theorem 5.4 can be relaxed to: R1 ′ .each graph in G is ℵ-small; R2 ′ .|G(x) ∩ G(y)| ≺ ℵ for every distinct x, y ∈ V (K ℵ ).
However, if ℵ is a singular cardinal, then conditions R1 ′ and R2 ′ are no longer sufficient.Indeed, we can construct a decomposition G of K ℵ into ℵ-small graphs such that a. |G| is ℵ-small, b.G satisfies conditions R1 ′ and R2 ′ , c. there are two (possibly isolated) vertices x and y belonging to every graphs of G, that is, G = G(x) ∩ G(y) Then, choosing any vertex z such that G(z) = G, we have that This means that condition N2 does not hold, therefore the decomposition G is not resolvable.
We conclude by showing that there is always a resolution for an 'almost' 2design with blocks that are ℵ ′ -bounded for some ℵ ′ ≺ ℵ, that is, a decomposition of K ℵ whose graphs are almost all ℵ ′ -bounded complete graphs.This extends some results on the resolvability of 2-designs given in [9].Proposition 5.6.Let G be a decomposition of the infinite complete graph K ℵ into ℵ ′ -bounded graphs for some ℵ ′ ≺ ℵ, where ℵ ′ is not necessarily infinite.If the subset of G consisting of all non-complete graphs is ℵ ′ -bounded, then G has a resolution.Proof.By assumption, condition R1 of Theorem 5.4 holds.To prove that G satisfies condition R2 for some ℵ ′′ ≺ ℵ, we assume for a contradiction the existence of vertices x and y such that |G(x) ∩ G(y)| ≻ ℵ ′′ := (ℵ ′ + 1).It follows that there are at least two complete graphs in G(x) ∩ G(y), meaning that the edge {x, y} is covered more than once by graphs in G, and this is a contradiction.The assertion follows from Theorem 5.4.

Theorem 1 .
4 allows us to construct decompositions of K ℵ into ℵ graphs of specified type.More precisely, we have the following.Corollary 5.1.Let F = {F α : α ∈ A} be an infinite family of (non-empty) ℵ-bounded graphs, where ℵ = |A|.Then there exists a decomposition