_{1}

^{*}

Let
D be a finite simple directed graph with vertex set
V(
D) and arc set
A(
D). A function
is called a signed dominating function (SDF) if
for each vertex
. The weight
of f is defined by
. The signed domination number of a digraph D is
. Let C_{m} × C_{n} denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of
g_{s}
(C_{m} × C_{n})
for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of
g_{s}
(C_{m} × C_{n})
when m,
(mod 3) and bounds for otherwise.

Throughout this paper, a digraph ^{+}, D^{−}). The minimum out-degree and in-degree of D are denoted by

graph D is

The Cartesian product _{1} and D_{2} is the digraph with vertex set

In the past few years, several types of domination problems in graphs had been studied [_{m} ´ C_{n} when m = 3, 4, 5, 6, 7 and arbitrary n. In this paper, we study the Cartesian product of two directed cycles C_{m} and C_{n} for mn ≥ 8n. We mainly determine the exact values of

Theorem 1.1 (Zelinka [

Lemma 1.2 (Zelinka [

Corollary 1.3 (Karami et al. [

In [

Theorem 1.4 [

In this section we calculate the signed domination number of the Cartesian product of two directed cycles C_{m} and C_{n} for m = 8, 9, 10 and

The vertices of a directed cycle C_{n} are always denoted by theintegers

Let us introduce a definition. Suppose that f is a signed dominating function for C_{m} ´ C_{n}, and assume that

Remark 2.1: Let f is a

Since C_{m} × C_{n} is 2-regular, it follows from

Remark 2.2. Since the case _{j} ≥ 0. Furthermore, s_{j} is odd if m is odd and even when m is even.

Let f be a signed dominating function for C_{m} ´ C_{n}, then we denote

column K_{j} and put

Then we have

For the remainder of this section, let f be a signed domination function of C_{m} × C_{n} with signed dominating sequence

Lemma 2.3. If

Proof. Let_{j} which get value −1. By Remark 2.1,

Theorem 2.4.

Proof. We define a signed dominating function f as follows:

By the definition of f, we have s_{j} = 2 for j is odd and s_{j} = 4 for j is even. Notice, f is a SDF for C_{8} × C_{n} when_{1} when

Now, let us define the following functions:

We note:

f_{1} is a SDF of C_{8} × C_{n} when

f_{2} is a SDF of C_{8} × C_{n} when

f_{3} is a SDF of C_{8} × C_{n} when

f_{4} is a SDF of C_{8} × C_{n} when

Hence,

For example, f_{1} is a SDF of C_{8} × C_{12}, where

{Here, we must note that, for simplicity of drawing the Cartesian products of two directed cycles C_{m} × C_{n}, we do not draw the arcs from vertices in last column to vertices in first column and the arcs from vertices in last row to vertices in first row. Also for each figure of C_{m} × C_{n}, we replace it by a corresponding matrix by signs − and + which descriptions −1 and +1 on figure of

By Remark 2.2, for any minimum signed dominating function f of C_{8} × C_{n} with signed dominating sequence_{j} = 0, 2, 4, 6 or 8 for_{j} = 0 then_{j} = 2 then

Hence, by (1), (2) and (3) we get

Assume that

Let f' ba a signed dominating function with signed dominating sequence

If m, n ≤ 7, then by Theorem 1.4 is the required (because

Claim 2.1. For k ≥ 2, we have

Proof of Claim 2.1. We have the subsequence

Now, if

Assume that

Case 1. If

Case 2. Let

For

Assume that

For the case 3, we need the following claim:

Claim 2.2. Let f' be a minimum signed dominating function of C_{8} × C_{n} with signed dominating sequence

Case 3. Let

Then we have

Since the case

If

If

Let

Then we have one possible is as the form_{8} × C_{n} when

By Lemma 1.2, and above arguments, we conclude that

Hence, from (1), (15) and (16), deduce that

Finally, we result that:

Theorem 2.5.

Proof. We define a signed dominating function f as follows:

By define f, we have s_{j} = 3 for_{9} × C_{n} for_{1} is a SDF of C_{9} × C_{n} for

From Corollary 1.3 is

For

If

By Remark 2.2, we have s_{j} = 1, 3, 5, 7 or 9. By Lemma 2.3, if s_{j} = 1 then_{j} = 3 then _{j} = 5 then _{j} ≥ 7). By Lemma 2.3, the

cases

We define

Then we have

If we have one case from the cases X_{9} ≥ 1, X_{7} ≥ 2, X_{5} + X_{7} ≥ 2 or X_{5} ≥ 3. Then by (19) is

Assume the contrary, i.e., (X_{9} = 0, X_{7} < 2, X_{5} + X_{7} < 2 and X_{5} < 3).

Hence,_{7} < 2 and X_{5} < 3, which are including the remained cases, i.e., X_{7} = 1 and X_{5} = 2. First, we give the following Claim:

Claim 2.3. There is only one possible for

The proof comes immediately by the drawing. □

Case 1. X_{7} = 1 and X_{5} = X_{9} = 0. Without loss of generality, we can assume s_{n} = 7. Then we have the form_{j}, _{j} and _{j}. Without loss of generality, we can assume

Subcase 1.1. For_{1}. This implies that

Subcase 1.2. For_{1}. This implies that

Case 2. X_{5} = 2 and_{j} = 3 otherwise. By the same argument similar to the Case 1, we have K_{j} is (j − 1)-shift of K_{1}. Thus, if_{1}, we always have

Subcase 2.1. d = 1, without loss of generality, we can assume

For_{n}, most including two values −1, and this is impossible. The same arguments is for

Subcase 2.2. d = 2, let

If n º 1(mod 3), then_{1}. Therefore, _{n} and this impossible. The same argument is for n º 2(mod 3).

Subcase 2.3. d = 3, let_{1}. Therefore_{n} with value −1. This is a contradiction, (because the vertices of the first column must be a signed dominates by the vertices of the last column). The same argument is for

Subcase 2.4. d ≥ 4, let

We have the form_{9} × C_{n} for

Finally, we deduce that does not exist a signed dominating function f of C_{9} × C_{n} for

From (18) and (20) is

Theorem 2.6.

Proof. We define a signed dominating function f as follows:

and

By define f and _{j} = 4 for all_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

is a SDF for C_{10} × C_{n} when

For an illustration

By Remark 2.2, we have s_{j} = 0, 2, 4, 6, 8 or 10. Also by Lemma 2.3, if s_{j} = 0, then _{j} = 2, is _{j} = 4 is _{j} ≥ 6). This implies that

So, we get

Corollary 2.7. For

Proof. By Corollary 1.3 we have

Let us a signed dominating function f as follows:

By define f, we have s_{j} = m/3 for_{m} × C_{n} for

For n º 1, 2(mod 3).

Let _{m} × C_{n} for

Thus,

This paper determined that exact value of the signed domination number of C_{m} × C_{n} for m = 8, 9, 10 and arbitrary n. By using same technique methods, our hope eventually lead to determination

Based on the above (Lemma 2.3 and Theorems 1.4, 2.4, 2.5 and 2.6), also by the technique which used in this paper, we again rewritten the following conjecture (This conjecture was mention in [

Conjecture 3.1.

RamyShaheen, (2015) On the Signed Domination Number of the Cartesian Product of Two Directed Cycles. Open Journal of Discrete Mathematics,05,54-64. doi: 10.4236/ojdm.2015.53005