#### Abstract

Some numerical characteristics of bipartite graphs in relation to the problem of finding all disjoint pairs of S-permutation matrices in the general case are discussed in this paper. All bipartite graphs of the type , where or , are provided. The cardinality of the sets of mutually disjoint S-permutation matrices in both the and cases is calculated.

#### 1. **Introduction**

Let be a positive integer. By we denote the set

We let denote the symmetric group of order , that is, the group of all one-to-one mappings of the set to itself. If , , then the image of the element in the mapping we will denote by .

A *bipartite graph* is an ordered triple
where and are nonempty sets such that . The elements of will be called *vertices*. The set of edges is . Multiple edges are not allowed in our considerations.

The subject of the present work is bipartite graphs considered up to isomorphism.

We refer to [1] or [2] for more details on graph theory.

Let and be two nonnegative integers, and let . We denote by the set of all bipartite graphs of the type , considered up to isomorphism, such that and .

Let , be square matrices, whose entries are elements of the set . The matrix
is called a *Sudoku matrix*, if every row, every column, and every submatrix , comprise a permutation of the elements of set , that is, every number is found just once in each row, column, and submatrix . Submatrices are called *blocks* of .

Sudoku is a very popular game, and Sudoku matrices are special cases of Latin squares in the class of gerechte designs [3].

A matrix is called *binary* if all of its elements are equal to 0 or 1. A square binary matrix is called *permutation matrix* if in every row and every column there is just one 1.

Let us denote by the set of all permutation matrices of the following type: where for every , is a square binary submatrix (block) with only one element equal to 1.

The elements of will be called *S-permutation matrices*.

Two matrices and , will be called *disjoint* if there are not elements and with the same indices such that .

The concept of S-permutation matrix was introduced by Dahl [4] in relation to the popular Sudoku puzzle.

Obviously, a square matrix with entries from is a Sudoku matrix if and only if there are matrices pairwise disjoint, such that can be written in the following way:

In [5] Fontana offers an algorithm which returns a random family of mutually disjoint S-permutation matrices, where . For , he ran the algorithm 1000 times and found 105 different families of nine mutually disjoint S-permutation matrices. Then, applying (1.5), he decided that there are at least Sudoku matrices. This number is very small compared with the exact number of Sudoku matrices. In [6] it was shown that there are exactly number of Sudoku matrices.

To evaluate the effectiveness of Fontana's algorithm, it is necessary to calculate the probability of two randomly generated matrices being disjoint. As is proved in [4], the number of S-permutation matrices is equal to

Thus the question of finding a formula for counting disjoint pairs of S-permutation matrices naturally arises. Such a formula is introduced and verified in [7]. In this paper, we demonstrate this formula to compute the number of disjoint pairs of S-permutation matrices in both the 4 4 and 9 9 cases.

#### 2. **A Formula for Counting Disjoint Pairs of S-Permutation Matrices**

Let for some natural numbers and , and let .

By we denote the set of all vertices of , adjacent with , that is, if and only if there is an edge in connecting and . If is an isolated vertex (i.e., there is no edge, incident with ), then by definition and . If , then obviously , and if , then .

Let , and let . We will say that and are equivalent, and we will write if . If and are isolated, then by definition if and only if and belong simultaneously to , or . The above introduced relation is obviously an equivalence relation.

By we denote the obtained factor set (the set of the equivalence classes) according to relation and let where , or , . We put and for every we define multiset (set with repetition) where are natural numbers, obtained by the above described way.

If is a permutation of the elements of the set and we shortly denote this permutation, then in this case we denote by the th element of this permutation, that is, .

The following theorem is proved in [7].

Theorem 2.1 (see [7]). *Let be a positive integer. Then the number of all disjoint ordered pairs of matrices in is equal to
*

The number of all nonordered pairs of disjoint matrices in is equal to

The proof of Theorem 2.1 is described in detail in [7], and here we will miss it.

In order to apply Theorem 2.1 it is necessary to describe all bipartite graphs up to isomorphism , where .

Let and are positive integers, and let . We examine the ordered -tuple where , is equal to the number of vertices of incident with exactly number of edges. It is obvious that is true for all . Then formula (2.4) can be presented as

Since and , then

Consequently, to apply formula (2.8) for each bipartite graph and for the set of bipartite graphs, it is necessary to obtain the following numerical characteristics:

Using the numerical characteristics (2.9), we obtain the following variety of Theorem 2.1.

Theorem 2.2. *One has
**
where is described using formulas (2.9).*

#### 3. **Demonstrations in Applying Theorem 2.2**

##### 3.1. **Counting the Number **** of All Ordered Pairs of Disjoint S-Permutation Matrices for **

###### 3.1.1. Consider

In and , consists of a single graph shown in Figure 1.

For graph we have

Then we get and therefore

###### 3.1.2. Consider

The set consists of three graphs , , and depicted in Figure 2.

For graph we have

For graphs and we have

Then for the set we get

###### 3.1.3. Consider

In and , consists of a single graph shown in Figure 3.

For graph we have

Then we get and therefore

###### 3.1.4. Consider

When and , there is only one graph, and this is the complete bipartite graph which is shown in Figure 4.

For graph we have

Then we get and therefore

Having in mind the formulas (2.10), (3.3), (3.6), (3.9), and (3.12) for the number of all ordered pairs disjoint S-permutation matrices in we finally get

The number of all nonordered pairs disjoint matrices from is equal to

##### 3.2. **Counting the Number **** of All Ordered Pairs of Disjoint S-Permutation Matrices for **

###### 3.2.1. Consider

Graph , which is displayed in Figure 5, is the only bipartite graph belonging to the set .

For graph we have

Then we get and therefore

###### 3.2.2. Consider

In this case . The graphs , , and are shown in Figure 6.

For graph we have

For graphs and we have

Then for the set we get

###### 3.2.3. Consider

When and , the set consists of six bipartite graphs, which are shown in Figure 7.

For graph we have

For graphs we have

For graph we have

For graphs we have

Then for the set we get

###### 3.2.4. Consider

When and , the set consists of seven bipartite graphs, which are shown in Figure 8.

For graph we have

For graph we have

For graph we have

For graphs and we have

For graphs and we have

Then we get

###### 3.2.5. Consider

When and , the set consists of seven bipartite graphs , which are shown in Figure 9.

For graph we have

For graph we have

For graph we have

For graphs and we have

For graphs and we have

Then we get

###### 3.2.6. Consider

When and , the set consists of six bipartite graphs, which are shown in Figure 10.

For graph we have

For graphs and we have

For graph we have

For graphs and we have

Then for the set we get

###### 3.2.7. Consider

When and the set consists of three bipartite graphs, which are shown in Figure 11.

For graph it is true that

For graphs and we get

Then for the set we get

###### 3.2.8. Consider

Graph , which is displayed in Figure 12, is the only bipartite graph belonging to the set in the case and .

For graph it is true that

Therefore,

###### 3.2.9. Consider

When and there is only one graph, and this is the complete bipartite graph which is shown in Figure 13.

For graph it is true that

Therefore

Having in mind the formula (2.10) and formulas (3.17) (3.49) for the number of all ordered pairs disjoint S-permutation matrices in we finally get

The number of all nonordered pairs disjoint matrices from is equal to

##### 3.3. **On a Combinatorial Problem of Graph Theory Related to the Number of Sudoku Matrices**

*Problem.* Let be a natural number, and let be a simple graph having vertices. Let each vertex of be identified with an element of the set of all S-permutation matrices. Two vertices are connected by an edge if and only if the corresponding matrices are disjoint. The problem is to find the number of all complete subgraphs of having vertices.

Note that the number of edges in graph is equal to and can be calculated using formulas (2.4) and (2.5) (resp., formulas (2.9), (2.10), and (2.5)).

Denote by the solution of the Problem 1, and let be the number of all Sudoku matrices. Then according to formula (1.5) and the method of construction of the graph , it follows that the next equality is valid:

We do not know a general formula for finding the number of all Sudoku matrices for each natural number , and we consider that this is an open combinatorial problem. Only some special cases are known. For example in it is known that [8]. Then according to formula (3.52) we get

In [6] it has been shown that in there are exactly, a number of Sudoku matrices. Then according to formula (3.52) we get