Abstract
A homogeneous factorisation of a graph Г is a partition of its arc set such that there exist vertex transitive subgroups M<G≤Aut(Г) such that M fixes each part setwise of the partition and Gpreserves the partition and transitive permuting the parts. In the present paper, we study homogeneous factorisation with M abelian. We give some interesting characterizations and constructions of such factorisations.
Introduction
If M is regular on VГ and MG  , then we call the factorisation an M-Cayley homogeneous factorisation; if, in particular, M is an abelian group, we occasionally simply call that Г has a homogeneous factorisation, M-Cayley homogeneous factorisation, M-abelian homogeneous factorisation and M-circulant homogeneous factorisation. The purpose of this paper is to character M-abelian homogeneous factorisation.
Introduction
The finite graphs homogeneous factorisation is initiated and researched by famous Algebra graphs theorem experts Praeger, Guralinck, and Saxl [1] . In 2003, Lim and Stringer gave characterizations for homogeneous factorisations of complete digraph and Edge-transitive homogeneous factorisations of complete graphs [2] , [3] . In 2004, Cuaresma studied homogeneous factorisations of Johnson graph [4] . In 2007, Giudici, Li, Potocnik, and Praeger accomplished homogeneous factorisations of complete multipartite graphs [5] . Fang, Li, and Wang characterized transitive 1-factorizations of arc-transitive graphs [6] . The general theory of homogeneous factorisation was studied in [1] , [7] . A necessary and sufficient condition for complete graphs having circulant homogeneous factorisations has been given by Praeger and Li, for complete graphs having (M,G) circulant homogeneous factorisations under the condition that G/M is a cyclic group [8] . This paper give a research in the base of the main work above.
Constructions
Let R be a group and
Constructions
with L a subgroup of 
Constructions
of Lemma 2.1. So we be sure that  has a factorisation whose factors are all of the same valency. Let
Constructions
Let M be a group with order bigger than 2. For a nonidentity automorphism  of M , the following construction gives a way to construct a M-Cayley cyclic ( , ) MG homogeneous factorisation
Constructions
 with 
Proof: let
( , ), ( ), ( ) 2. Cay M S Aut M o n      
Claim 1:
, xS  we have
Claim 1:
According to Claim 1, we know S is the union set of some ⟨   -orbits in M and 
Claim 1:
obviously has no homogeneous factorisation, which proves the necessity of the proposition. In lemma 3.1. We prove that complete graph of prime power vertices has abelian homogeneous factorisation. This leads to the following question. The following proposition give a sufficient condition for the existence of abelian homogeneous factorisation of the complete graphs with not prime power vertices, thus provides a partial answer of Question 3.3. 
Abelian Homogeneous Factorisation of Complete Graphs
the Singer subgroup of (2, ) : 
International Journal of Applied Physics and Mathematics

Abelian homogeneous factorization of some complete multipartite graphs
o o x o y     Construction 4.2. ( ) , ( ) 1,( , 1) 1 o p o q p q        Claim:   2 (1) pq p K  or   1 ( , )( , ) ( , )u q q GL q       . Let ( , ), , x y x y         , ( ) ( ) ( ) o o x o y p     1 (2, ) 01 pp b x b F GL p                    1 1 10 10 0 \ {0} 0 0 q p q q p y d d F d d                         

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