aa r X i v : . [ m a t h . C O ] O c t Abelian 1 − factorizations of complete multipartitegraphs M. Bogaerts
June 3, 2018
Abstract
An automorphism group G of a 1 − factorization of the complete multi-partite graph K m × n consists in permutations of the vertices of the graphmapping factors to factors. In this paper, we give a complete answer tothe existence or non-existence problem of a 1 − factorization of K m × n ad-mitting an abelian group acting sharply transitively on the vertices of thegraph. In this paper, we consider 1 − factorizations of complete multipartite graphs K m × n (i.e. with m parts consisting of n vertices). In order to avoid confusionwith the complete graph K m = K m × , it is from now assumed that each part of K m × n has at least two vertices (i.e. n ≥ r − factor of a graph Γ is a span-ning subgraph with all vertices of degree r . For r = 1, the concept of 1 − factorcoincide with the definition of a perfect matching. An r − factorization of agraph Γ is a decomposition of Γ as a union of disjoint r − factors.A group of permutations of the vertices of Γ mapping factors to factors is calledan automorphism group of the r − factorization . If the action of an auto-morphism group is sharply transitive on the vertices, the factorization is said tobe sharply transitive .In [6], G. Mazzuoccolo and G. Rinaldi consider the following problem: Problem 1.
Given a finite group G of even order, which graphs Γ have a − factorization admitting G as an automorphism group with a sharply transitiveaction on the vertex set? The problem above is the natural generalization of the largely studied casein which we set Γ to be the complete graph. The benchmark on this topic isthe following theorem of Hartman and Rosa [5] from 1985:
Theorem 1.
A complete graph K n admits a − factorization with a cyclic au-tomorphism group acting sharply transitively on the vertices if and only if n iseven and n = 2 t , t ≥ . − factorization of the complete multipartite graph admitting acyclic automorphism group acting sharply transitively on the vertex-set (a cyclic1-factorization from now on). Theorem 2. [1, 6] A cyclic − factorization of K m × n : • does not exist if m ≡ , n = 2 d where d is odd • does not exist if m = 2 v d n = 2 d ′ where d and d ′ are odd and v ≥ • does not exist if m = p v , n = 2 , where p is prime such that p ≡ and v ≥ , • exists if m = 2 d , n = 2 d ′ where d and d ′ are odd. • exists if m = 2 v d , n = 2 u d ′ where d and d ′ are odd and u > . • exists if m = 2 v d , n = d ′ where d and d ′ are odd and v ≥ . • exists if n = 2 , m ≡ and m is not a prime power, • exists if m ≡ and n = 2 d where d > is odd. In this paper, we consider the following extension of problem 1:
Problem 2.
Given a graph Γ , does a − factorization of Γ admitting an abeliangroup G as automorphism group with a sharply transitive action on the vertexset exists? Of course, if a cyclic factorization exists, then this question has a positiveanswer. On the other hand, the trivial condition mn ≡ Theorem 3.
An abelian − factorization of K m × n : • exists if m = 2 v d n = 2 d ′ where d and d ′ are odd and v ≥ , • does not exist if m = p , n = 2 , where p is prime such that p ≡ , • exists if m = p v , n = 2 , where p is prime such that p ≡ and v ≥ , • does not exist if m ≡ , n = 2 d where d is odd The material involved in the constructions is presented in section 2. Sections3 to 5 state the existence and non-existence results.2
Preliminaries
This paper is dedicated to the construction of 1 − factorization of K m × n ( mn even) with an abelian automorphism group G acting sharply transitively on thevertices. For sake of clarity, the composition is written additively, with neutralelement ¯0. As in [6], the graph Γ = K m × n is considered as the Cayley graph Cay ( G, Ω) with Ω = G − H where G is a group of order mn and H is a subgroupof order n of G . Vertices of Γ are the elements of G , and edges are (unordered)pairs of elements [ g , g ] , g i ∈ G such that the difference g − g belongs to Ω.With this definition, the group G acts naturally sharply transitively on its ele-ments (i.e. on the vertices of Γ).In [6], G. Mazzuoccolo and G. Rinaldi give the definition of a starter for K m × n , a generalization of the concept defined by M. Buratti in [4]. The exis-tence of a starter is proven to be equivalent to the existence of a 1 − factorizationof K m × n with an automorphism group G acting sharply transitively on the ver-tices. We here give the definition of a starter for K m × n with a abelian group.The set Ω can be partitioned as Ω ∪ Ω ∪ Ω ′ where Ω contains all involu-tions of Ω, and for any g ∈ Ω , − g ∈ Ω ′ . The edge set of Γ can be describedas the union of orbits of G acting additively, i.e. ( ∪ j ∈ Ω { [ k, j + k ] : k ∈ G } )) ∪ ( ∪ g ∈ Ω { [ l, l + g ] : l ∈ G } )). For any j ∈ Ω , Orb G ([¯0 , j ]) = { [ k, j + k ] : k ∈ G } has mn elements and forms a 1 − factor of Γ. Because there are only mn elementsin each orbit, edges of this kind are called short edges .If g ∈ Ω , then Orb G ([¯0 , g ]) has 2 n elements and is a union of cycles of Γ.These cycles can be written as ( x, x + g, x + 2 g, . . . , x + ( k − g ) where k is theorder of g and x is a representative of one right coset of < g > in G . The edgesin these orbits are called long edges .Now we define two mappings ∂ , φ . The first one maps an edge to thedifferences of its vertices, and the second one maps an edge to its vertices. Bothgive one element if the edge is short, and 2 if the edge is long. ∂ ([ x, y ]) = (cid:26) { x − y, y − x } if [ x, y ] is long { x − y } if [ x, y ] is short φ ([ x, y ]) = (cid:26) { x, y } if [ x, y ] is long { x } if [ x, y ] is shortThen for a set S , we define ∂ ( S ) = ∪ e ∈ S ∂ ( e ) and φ ( S ) = ∪ e ∈ S φ ( e ).Now we give the definition of a starter for the pair ( G, Ω): it is a set Σ = { S , . . . , S k } of subsets of the edges E (Γ), with k subgroups H , . . . , H k of G such that: • the union of the differences ∂S ∪ · · · ∪ ∂S k is Ω, but without repetition(every element appears exactly once).3 for each S i , φ ( S i ) contains exactly one representative of each right cosetof H i in G . • for any H i and any short edge [ x, y ] ∈ S i , H i contains the involution y − x = x − y .Each H i − orbit of a set S i ∈ Σ is a 1 − factor of Γ and the action of G on the1 − factors covers all edges of Γ exactly once. By construction, the factorizationis preserved under the action of G .The construction of a starter can be simplified by the existence of a subgroup A of G of index 2, as mentioned in [6]: Proposition 4. [6] Let G be a finite group possessing a subgroup A of index2 and let Σ ′ = { S , . . . , S t } be a set of subsets of E (Γ) together with subgroups H , . . . , H t which satisfy the second and third conditions of the definition of astarter. If ∂S ∪ · · · ∪ ∂S t ⊂ A ∩ Ω and it does not contain repeated elements,then Σ ′ can be completed to a starter for the pair ( G, Ω) . -factorization of K v d × d ′ , d, d ′ odd, v ≥ First, consider the case d ′ = 1. In [2], Bonisoli and Labbate have proven thefollowing proposition: Proposition 5.
For m = 2 v d , v ≥ , d odd, there exists a − factorization of K m with C m × Z as vertex-regular automorphism group, with one fixed factor F . The graph obtained by deletion of the edges of F is K m × , proving that thisgraph has a 1 − factorization preserved by a sharply transitive abelian automor-phism group. We generalize this construction for other values as follows. Proposition 6.
For m = 2 v d , n = 2 d ′ , v ≥ , d and d ′ odd, there exists a − factorization of K m × n with G = C d × C v d ′ × Z as vertex-regular automor-phism group. The proof of this proposition relies on the following construction lemma.
Lemma 7.
If there exists a starter for a cyclic regular − factorization of K m × n with automorphism group G , then there exists a starter for K m × n with G ′ = G × Z . Proof of lemmma 7
By hypothesis, K m × n has a starter for the cyclic group G = Z mn with sub-group H of order n . Let Σ = { S , . . . , S k } together with subgroups H , . . . , H k
4e this starter. Now let consider G ′ = G × Z with subgroup H ′ = H × Z , insuch a way that Ω ′ = G ′ \ H ′ give K m × n = Cay ( G ′ , Ω ′ ).Let Σ ′ denote the starter constructed with sets { S , . . . , S k , S , . . . , S k } withsubgroups H , . . . , H k , H , . . . , H k . For i = 1 , . . . , k , S i (resp. S i ) is definedas the set of pairs [( a, , ( b, a, , ( b, a, b ] in S i , and H i = H i is defined as H i × Z .Elements in Ω ′ = G ′ \ H ′ can be written as ( a, c ) with a ∈ Ω and c ∈ Z .By construction, the union δS ∪ · · · ∪ δS k covers exactly once all elements oftype ( a,
0) with a ∈ Ω, while the union δS ∪ · · · ∪ δS k does the same with allelements of type ( a,
1) with a ∈ Ω. (cid:3) Since a cyclic regular 1 − factorization of K v d × d ′ always exists, as stated intheorem 2, proposition 6 is easily deduced from lemma 7. -factorization of K p v × , p ≡ prime, v ≥ The case K p v d × , p ≡ v ≥ v = 1, up to isomorphism, the only possible abelian group isthe cyclic group of order 2 p , for which it is known that a starter does not ex-ists. For all v ≥
2, a starter always exists, as stated in the following proposition.
Proposition 8.
For m = p v , p ≡ prime , v ≥ , there exists a − factorization of K m × with G = Z p v − × Z p × Z as vertex-regular automor-phism group. Proof
The existence of the 1 − factorization will follow from the constuction of thestarter. First, the subgroup H of G = Z p v − × Z p × Z is defined by its generator(0 , , p = 4 t + 1 and t ′ = p v − − , and let define the pairs ( S i , H i ).Let H = < (1 , , > . The set S is defined as the union ( ∪ j =1 S j ) ∪ { e , e } where: S = { [(0 , i, , (0 , p − i, i = 1 , . . . , t } , S = { [(0 , i − , , (0 , p − i, i = 1 , . . . , t } , S = { [(0 , t + i, , (0 , p − t − i, i = 1 , . . . , t } , S = { [(0 , t + i + 1 , , (0 , p − t − i, i = 1 , . . . , t − } , e = [(0 , , , (2 , t, e = [(0 , t + 1 , , (2 , t + 1 , H m is defined for m = 1 , . . . , t ′ by: H m = H , S m = { [(0 , i, , (2 m − , p − i, i = 1 , . . . t } ,5 m = { [(0 , i − , , (2 m − , p − i, i = 1 , . . . t , S m = { [(0 , p − t − i, , (2 m, t + i, i = 1 , . . . t } , S m = { [(0 , p − t − i, , (2 m, t + i − , i = 1 , . . . t } , S m = { [(0 , , , (2 m − , t, } For each m , S m is defined by the union ∪ j =1 S jm Now let H t ′ +1 = < (0 , , > and define S t ′ +1 := ∪ j =1 S j t ′ +1 with thefollowing subsets. S t ′ +1 = { [(1 − i, , , ( i, , i = 1 , . . . , t ′ } , S t ′ +1 = { [( i, , , ( − i, , i = 1 , . . . , t ′ } , S t ′ +1 = { [( t ′ + i, , ) , ( − t ′ − i, t + 2 , i = 1 , . . . , t ′ } , S t ′ +1 = { [(0 , , , ( − t ′ , , } Let A be the subgroup Z p v − × Z p of index 2 of G . Let Σ ′ = { S , . . . , S t ′ +1 } be the family of subsets of edges together with subgroups ( H , . . . , H t ′ +1 ). Itis a simple check that all elements ( i, j, k ) of Ω with k = 0 are covered by δ ( ∪ l S l ), and that the second and third conditions of the definition of a starterare satisfied. Proposition 4 can be applied, resulting in the existence of theappropriate starter. (cid:3) -factorization of K m × d , m ≡ , d odd Proposition 9.
For m ≡ , d odd, an abelian − factorization of K m × d with a sharply transitive action on the vertices does not exist. Proof
Here we use the well-known fact that any abelian group G can be written asa direct product of cyclic group whose orders are equal to a power of a prime: G = C p v × · · · × C p vkk , p i prime for all i = 1 , . . . , k . Supose that there existsa starter for an abelian regular 1 − factorization of K m × d . The group G musthave order 2 md , and is isomorphic to the direct product G ′ × Z , where G ′ the abelian subgroup of index 2. The subgroup H has order 2 d and can bedecomposed as H = H ′ × Z where H ′ is an abelian group order d . The setΩ = G \ H has 2 md − d = 2 d ( m −
1) elements, none of them is an involution.The number of elements in Ω of type ( a,
0) is equal to the number of elementsthat can be written as ( a, H k choosen to construct the starter are of odd order, so the set φ ( S k ) of representative of the cosets counts as many elements of type ( a,
0) thanelements of type ( b, e in S k with δe = { ( a, , ( − a, } is even. Each of these edges contributes to covertwo elements of Ω with δe . For each k , δS k covers 4 r elements of type ( a,
0) ofΩ. On the other hand, Ω contains d ( m − ≡ (cid:3) eferences [1] Mathieu Bogaerts and Giuseppe Mazzuoccolo, Cyclic and dihedral − factorizations of multipartite graphs , The Electronic Journal of Combi-natorics (2011), no. P179.[2] Arrigo Bonisoli and Domenico Labbate, One-factorization of completegraphs with vertex-regular automorphism groups , Journal of CombinatorialDesigns (2002), 1–16.[3] Simona Bonvincini and Giuseppe Mazzuoccolo, Abelian one-factorizations ininfinite graphs , European Journal of Combinatorics (2010), 1847–1852.[4] Marco Buratti, Abelian 1-factorizations of the complete graph , EuropeanJournal of Combinatorics (2001), 291–295.[5] Alan Hartman and Alexander Rosa, Cyclic one-factorization of the completegraph , European Journal of Combinatorics (1985), 45–58.[6] Giuseppe Mazzuoccolo and Gloria Rinaldi, Sharply transitive − factorization of complete multipartite graphs , The Electronic Jour-nal of Combinatorics17