The present paper considers multipartite graphs from the perspective of design theory and coding theory. A one-factor $F$ of the complete multipartite graph $K_{n\times g}$ (with $n$ parts of size $g$) gives rise to a $(g+1)$-ary code ${\cal C}$ of length $n$ and constant weight two. Furthermore, if the one-factor $F$ meets a certain constraint, then ${\cal C}$ becomes an optimal code with minimum distance three. We initiate the study of one-factorizations of complete multipartite graphs subject to distance constraints. The problem of decomposing $K_{n\times g}$ into the largest subgraphs with minimum distance three is investigated. It is proved that, for $n\le g$, the complete multipartite graph $K_{n\times g}$ can be decomposed into $g^2$ copies of the largest subgraphs with minimum distance three. For even $gn$ with $n>g$, it is proved that the complete multipartite graph $K_{n\times g}$ can be decomposed into $g(n-1)$ one-factors with minimum distance three, leaving a small gap of $n$ (in terms of $g$) to be resolved (If $gn$ is odd when $n>g$, no such decomposition of $K_{n\times g}$ exists).
One-factors and one-factorizations of graphs arise naturally in tournament applications and they occur as building blocks in many combinatorial designs and structures. The relationships to Steiner triple systems and various Latin squares are canonical topics which stirred up research interest in one-factorizations [1,5,8,13,14]. Special types of one-factorizations have been studied extensively, such as cyclic, perfect, indecomposable, and orthogonal one-factorizations [15]. This paper will consider one-factorizations of complete multipartite graphs subject to distance constraints.
We consider decomposition of multipartite graphs from the perspective of constant-weight codes. A one-factor F of a complete multipartite graph K n×g (with n parts of size g) gives rise to a (g + 1)-ary code C of length n and constant weight two. Furthermore, if the one-factor F meets a certain constraint, then C becomes an optimal code with minimum distance three. Similarly we will have that a one-factorization of K n×g is a partition of the set of all words of length n with weight two over an alphabet of size q = g + 1 into optimal q-ary codes with minimum distance three.
A multipartite graph, or an n-partite graph, is a graph with vertices partitioned into n parts such that no two vertices within the same part are adjacent. If n = 2, this defines a bipartite graph. A complete multipartite graph is a multipartite graph such that every pair of vertices in two different parts are adjacent. The complete multipartite graph with n parts of size g is denoted by K n×g . By this definition, the complete graph K n is K n×1 .
A graph G is r-regular if every vertex of G has degree r. It is almost r-regular if every vertex of G has degree r or r -1. A one-factor is a 1-regular subgraph of G. A one-factorization of G is a decomposition of the edge set of G into edge-disjoint one-factors. A near one-factor of G is a subgraph, which has one isolated vertex and all other vertices have degree 1. A set of near one-factors which covers every edge of G precisely once is called a near one-factorization of G. For simplicity, we usually denote a graph solely by its edge set. Specifically, the interval [1, n] is abbreviated as [n] for n ∈ Z + . For a complete graph K 2n , take the set of vertices to be [2n -1] ∪ {∞}. For j ∈ [2n -1], define F j = {∞, j}, {j + 1, j -1} , {j + 2, j -2} , . . . , {j + n -1, j -n + 1} , where j+i and j-i (i ∈ [n-1]) are reduced by modulo 2n-1 to lie in [2n -1]. It is not difficult to see that F 1 , F 2 , . . . , F 2n-1 forms a one-factorization of K 2n . A near one-factorization of K 2n-1 can be constructed from a one-factorizations of K 2n by deleting a single vertex and its incident edges from K 2n . For the complete multipartite graph K n×g , it is also well-known that there exists a one-factorization if and only if gn is even [10].
For a positive integer q, denote by Z q the additive group of integers modulo q. Let Z n q be the set of all words of length n over the alphabet Z q . A word x ∈ Z n q is denoted by x = (x 1 , x 2 , . . . , x n ). The (Hamming) distance between two words x and y is defined as:
The weight of a word x is the number of nonzero entries in x, i.e., d(x, 0). The support of x, denoted by supp(x), is the set of coordinate positions i ∈ [n] whose entry x i is nonzero.
Let H q (n, w) denote the set of all words of length n and weight w over Z q . A q-ary constantweight code C of length n, weight w and distance d, denoted (n, d, w) q -code, is a nonempty subset of H q (n, w) such that d(x, y) ≥ d for all distinct x, y ∈ C. Every element of C is called a codeword. The maximum size of an (n, d, w) q -code is denoted A q (n, d, w). An optimal (n, d, w) qcode is an (n, d, w) q -code having A q (n, d, w) codewords. The fundamental problem in coding theory is that of determining A q (n, d, w) and the relevant values in our exposition are as follows. Throughout the paper, we assume that q = g + 1.
Lemma 1.1 ([9]).
(1) A q (n, 2, w) = n w g w-1 .
(2) A q (n, 2w, w) = ⌊ n w ⌋.
A q (n, 3, 2) = min
Throughout the paper, let all n-partite graphs be defined over the vertex set V = [n] × [g] with the n parts P x = {x} × [g], x ∈ [n], unless otherwise stated. If arithmetic operations are taken in the expression of a vertex in V , then note that the result lies in V , usually with the operation for the first component reduced modulo n to lie in [n] and the second reduced modulo g to lie in [g], unless otherwise specified. Each edge e ∈ K n×g , consisting of two vertices (x, a), (y, b), defines a word c e ∈ H q (n, 2), whose x-th coordinate is a, y-th coordinate is b, and all other coordinates are zeros. As a result, an n-partite graph G is equivalent to a constant-weight code C which contains certain 2-weight words. This establishes a one-to-one correspondence between multipartite graphs G and q-ary codes C of constant weight two. The multipartite graph G is said to have distance d if its associated code C
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