A Note on Decomposition of Tensor Product of Complete Multipartite Graphs into Gregarious Kite

Elakkiya, A. Tamil

doi:10.61091/jcmcc120-26

Abstract

1. Introduction
2. Preliminary Results
3. Existence of GK-decomposition of (Km⊗K¯n)×(Kr⊗K¯s)
4. Main Result
5. Conclusion
Funding Information
Conflict of Interest

References

Journal of Combinatorial Mathematics and Combinatorial Computing

Volume 120
Pages: 295-299

Research article

A Note on Decomposition of Tensor Product of Complete Multipartite Graphs into Gregarious Kite

^¹

¹Gobi Arts & Science College, Gobichettipalayam, Erode, Tamil Nadu, India

Received: 19/01/2024
Accepted: 22/06/2024
Published: 30/06/2024

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Abstract

A kite $K$ is a graph which can be obtained by joining an edge to any vertex of $K_{3}$ . A kite with edge set ${a b, b c, c a, c d}$ can be denoted as $(a, b, c; c d)$ . If every vertex of a kite in the decomposition lies in different partite sets, then we say that a kite decomposition of a multipartite graph is a gregarious kite decomposition. In this manuscript, it is shown that there exists a decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ into gregarious kites if and only if
$n^{2} s^{2} m (m - 1) r (r - 1) \equiv 0 (\mod 8),$
where $\otimes$ and $\times$ denote the wreath product and tensor product of graphs respectively. We denote a gregarious kite decomposition as $G K$ -decomposition.

Keywords: Decomposition, Kite, Tensor product, Wreath Product

1. Introduction

Many authors have been studied on kite-design and kite decomposition/ kite factorization and their special properties such as gregarious kite decomposition/ gregarious kite factorization. Bermond and Schonheim [1] proved that a kite-design of order $n$ exists if and only if $n \equiv 0, 1 (\mod 8)$ . Wang and Chang [2,3] considered the problem of a resolvable $(K_{3} + e)$ and $(K_{3} + e, λ)$ -group divisible designs of type $g^{t} u^{1} .$ Wang [4] have proven that the necessary conditions to find a resolvable $(K_{3} + e)$ -group divisible design of type $g^{u}$ are also sufficient. Fu et al. [5] have demonstrated that there exists a $G K$ -decomposition of $K_{m} (n)$ if and only if $n \equiv 0, 1 (\mod 8)$ for odd $m$ or $n \geq 4$ for even $m .$ Gionfriddo and Milici [6] considered the uniformly resolvable decompositions of $K_{v}$ and $K_{v} - I$ into paths and kites. To know more about kite designs, refer [7-11]. Further, the authors A. Tamil Elakkiya and A. Muthusamy [12] have shown that there is a $G K$ -decomposition of $K_{m} \times K_{n}$ if and only if $m n (m - 1) (n - 1) \equiv 0 (\mod 8) .$ Moreover, the authors A. Tamil Elakkiya and A. Muthusamy [13] considered the existence of a $G K$ factorization of tensor product of complete graphs.

In this way, our main concern is to find a existence of a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ . In this document, it is demonstrated that a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ holds, if and only if, $n^{2} s^{2} m (m - 1) r (r - 1) \equiv 0 (\mod 8) .$

2. Preliminary Results

Definition 1. [12] A partition of a graph $G$ into a collection of subgraphs $G_{1}, G_{2}, \dots, G_{r}$ such that each one of it is distinct mutually by their edges together with $E (G) = \cup_{i = 1}^{r} E (G_{i})$ is called a decomposition of $G;$ We then write $G$ as $G = G_{1} \oplus G_{2} \oplus \dots \oplus G_{r},$ where $\oplus$ denotes edge-disjoint sum of subgraphs. For an integer $s,$ $s G$ denotes $s$ copies of $G .$

Definition 2. [12] A kite $K$ is a graph which can be obtained by joining an edge to any vertex of $K_{3}$ , for instance see Figure 1. A kite with edge set ${a b, b c, c a, c d}$ can be denoted as $(a, b, c; c d)$ .

If every vertex of a kite in the decomposition lies in different partite sets, then we say that a kite decomposition of a multipartite graph is a gregarious kite decomposition. Here we may consider a Gregarious Kite decomposition as a $G K$ -decomposition for short.

Definition 3. [12] The tensor product $G \times H$ and the wreath product $G \otimes H$ of two graphs $G$ and $H$ are defined as follows: $V (G \times H)$ = $V (G \otimes H)$ = ${(u, v) ∣ u \in V (G), v \in V (H)} .$ $E (G \times H)$ = ${(u, v) (x, y) ∣ u x \in E (G)$ and $v y \in E (H)}$ and $E (G \otimes H)$ = ${(u, v) (x, y) ∣ u = x$ and $v y \in E (H)$ or $u x \in E (G)} .$

As tensor product has commutative and distributive property over an edge-disjoint sum of subgraphs, if decomposition of $G$ = $G_{1} \oplus G_{2} \oplus \dots \oplus G_{r}$ holds, we then write as $G \times H$ = $(G_{1} \times H) \oplus (G_{2} \times H) \oplus \dots \oplus (G_{r} \times H) .$

Definition 4. [12] A graph $G$ having partite sets $V_{1}, V_{2}, \dots, V_{m}$ with $∣ V_{i} ∣= n,$ $1 \leq i \leq n$ and $E (G)$ ={ $u v$ $∣$ $u \in V_{i}$ and $v \in V_{j}, \forall i \neq j$ } is called complete $m$ -partite graph and is denoted by $K_{m} (n) .$ Note that $K_{m} (n)$ is same as the $K_{m} \otimes {\bar{K}}_{n} .$ where ${\bar{K}}_{n}$ is the complement of a complete graph on $n$ vertices.

In order to prove our key result, we require the following:

Theorem 1. [5] There exists a $G K$ -decomposition of $K_{m} (n)$ if and only if $m \equiv 0, 1 (\mod 8),$ for all odd $n$ (or) $m \geq 4,$ for even $n .$

Remark 1. [5] Let $K$ is a kite. Then there exists a $G K$ -decomposition of $K \otimes {\bar{K}}_{s},$ for all $s .$

Lemma 1. [12] For all $n,$ $n \neq 2$ , there exists a $G K$ -decomposition of $K \times K_{n},$ where $K$ is a kite.

Theorem 2. [12] There exists a $G K$ -decomposition of $K_{m} \times K_{n}$ if and only if $m n (m - 1) (n - 1) \equiv 0 (\mod 8) .$

3. Existence of $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$

Lemma 2. A $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ exists, if $m \equiv 0, 1 (\mod 8),$ $n, s \equiv 1 (\mod 2)$ and for all $r,$ $r \neq 2.$

Proof. By Theorem 1, we have a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) .$ Then $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ = $(K \oplus K \oplus \dots \oplus K) \times (K_{r} \otimes {\bar{K}}_{s})$ = $K \times (K_{r} \otimes {\bar{K}}_{s}) \oplus K \times (K_{r} \otimes {\bar{K}}_{s}) \oplus \dots \oplus K \times (K_{r} \otimes {\bar{K}}_{s}) .$ Now $K \times (K_{r} \otimes {\bar{K}}_{s}) ≅ (K \times K_{r}) \otimes {\bar{K}}_{s} .$ As we have a kite decomposition of $K \times K_{r}$ from Lemma 1, we then write as $(K \times K_{r}) \otimes {\bar{K}}_{s}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{s}$ = $(K \otimes {\bar{K}}_{s}) \oplus (K \otimes {\bar{K}}_{s}) \oplus \dots \oplus (K \otimes {\bar{K}}_{s}) .$ A $G K$ -decomposition of $K \otimes {\bar{K}}_{s}$ can be obtained from Remark 1. Thus all together provides a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) .$

Lemma 3. A $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ exists, if $n \equiv 0 (\mod 2)$ and for all $m, r, s,$ ${m, r} \neq 2.$

Proof. Case 1: Let $m = 3,$ $n = 2 b, b \geq 0$ and for all $r, s, (r \neq 2) .$ We write $(K_{3} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) ≅ [(K_{3} \otimes {\bar{K}}_{n}) \times K_{r}] \otimes {\bar{K}}_{s}$ = $[(K_{3} \oplus K_{3} \oplus \dots \oplus K_{3}) \times K_{r}] \otimes {\bar{K}}_{s}$ = $[(K_{3} \times K_{r}) \oplus (K_{3} \times K_{r}) \oplus \dots \oplus (K_{3} \times K_{r})] \otimes {\bar{K}}_{s}$ = $[(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}] \oplus [(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}] \oplus \dots \oplus [(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}] .$ Theorem 2 gives a $G K$ -decomposition for $K_{3} \times K_{r}$ . Now, we write as $(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{s}$ = $(K \otimes {\bar{K}}_{s}) \oplus (K \otimes {\bar{K}}_{s}) \oplus \dots \oplus (K \otimes {\bar{K}}_{s}) .$ A $G K$ -decomposition exists for $K \otimes {\bar{K}}_{s}$ according to Remark 1. As a consequence of it, a $G K$ -decomposition of $(K_{3} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ has been derived.

Case 2: A $G K$ -decomposition for $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ have been described as follows:

Let $m \geq 4,$ $n = 2 b, b \geq 0$ and for all $r, s, (r \neq 2) .$ One can obtain a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n})$ from Theorem 1. Then $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ = $(K \oplus K \oplus \dots \oplus K) \times (K_{r} \otimes {\bar{K}}_{s})$ = $K \times (K_{r} \otimes {\bar{K}}_{s}) \oplus K \times (K_{r} \otimes {\bar{K}}_{s}) \oplus \dots \oplus K \times (K_{r} \otimes {\bar{K}}_{s}) .$ Now $K \times (K_{r} \otimes {\bar{K}}_{s}) ≅ (K \times K_{r}) \otimes {\bar{K}}_{s} .$ A kite decomposition exits for $K \times K_{r}$ by using Lemma 1. Moreover, we then write, $(K \times K_{r}) \otimes {\bar{K}}_{s}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{s}$ = $(K \otimes {\bar{K}}_{s}) \oplus (K \otimes {\bar{K}}_{s}) \oplus \dots \oplus (K \otimes {\bar{K}}_{s}) .$ According to Remark 1, a $G K$ -decomposition of $K \otimes {\bar{K}}_{s}$ has been proved.

Consequently, Cases $1$ and $2$ lead a $G K$ -decomposition for $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ when $n \equiv 0 (\mod 2)$ and for all $m, r, s,$ ${m, r} \neq 2.$

Lemma 4. A $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ exists, if $n \equiv 1 (\mod 2),$ $s \equiv 0 (\mod 2),$ and for all $m, r,$ ${m, r} \neq 2.$

Proof. Case 1: Let $m = 3,$ $n = 2 b + 1, b \geq 0,$ $s = 2 c, c \geq 0$ and for all $r, (r \neq 2) .$ Let us consider $(K_{3} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) ≅ [(K_{3} \otimes {\bar{K}}_{n}) \times K_{r}] \otimes {\bar{K}}_{s}$ = $[(K_{3} \oplus K_{3} \oplus \dots \oplus K_{3}) \times K_{r}] \otimes {\bar{K}}_{s}$ = $[(K_{3} \times K_{r}) \oplus (K_{3} \times K_{r}) \oplus \dots \oplus (K_{3} \times K_{r})] \otimes {\bar{K}}_{s}$ = $[(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}] \oplus [(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}] \oplus \dots \oplus [(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}] .$ By Theorem 2, a $G K$ -decomposition exists for $K_{3} \times K_{r}$ . Now $(K_{3} \times K_{r}) \otimes {\bar{K}}_{s}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{s}$ = $(K \otimes {\bar{K}}_{s}) \oplus (K \otimes {\bar{K}}_{s}) \oplus \dots \oplus (K \otimes {\bar{K}}_{s}) .$ Now, a $G K$ -decomposition of $K \otimes {\bar{K}}_{s}$ follows from Remark 1. Thus, the above construction provides a $G K$ -decomposition of $(K_{3} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ .

Case 2: Let $m \geq 4,$ $n = 2 b + 1, b \geq 0,$ $s = 2 c, c \geq 0$ and for all $r, (r \neq 2) .$ By Theorem 1, there is a $G K$ -decomposition for $(K_{r} \otimes {\bar{K}}_{s})$ . Then $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ = $(K_{m} \otimes {\bar{K}}_{n}) \times (K \oplus K \oplus \dots \oplus K)$ = $[(K_{m} \otimes {\bar{K}}_{n}) \times K] \oplus [(K_{m} \otimes {\bar{K}}_{n}) \times K] \oplus \dots \oplus [(K_{m} \otimes {\bar{K}}_{n}) \times K] .$

Now $[(K_{m} \otimes {\bar{K}}_{n}) \times K] ≅ K \times (K_{m} \otimes {\bar{K}}_{n}) ≅ (K \times K_{m}) \otimes {\bar{K}}_{n} .$ By Lemma 1, we have a kite decomposition of $K \times K_{m} .$ Then $(K \times K_{m}) \otimes {\bar{K}}_{n}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{n}$ = $(K \otimes {\bar{K}}_{n}) \oplus (K \otimes {\bar{K}}_{n}) \oplus \dots \oplus (K \otimes {\bar{K}}_{n}) .$ A $G K$ -decomposition of $K \otimes {\bar{K}}_{n}$ follows from Remark 1. Thus, all together gives a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) .$

Lemma 5. A $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ exists, if $m \equiv 4, 5 (\mod 8),$ $n, s \equiv 1 (\mod 2)$ and for all $r,$ $r \neq 2.$

Proof. Let us consider $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) ≅ [(K_{m} \otimes {\bar{K}}_{n}) \times K_{r}] \otimes {\bar{K}}_{s} .$ We write as $(K_{m} \otimes {\bar{K}}_{n}) \times K_{r} ≅ K_{r} \times (K_{m} \otimes {\bar{K}}_{n}) ≅ (K_{r} \times K_{m}) \otimes {\bar{K}}_{n} .$ By Theorem 2, we have a $G K$ -decomposition of $K_{r} \times K_{m} .$ Then $(K_{r} \times K_{m}) \otimes {\bar{K}}_{n}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{n}$ = $(K \otimes {\bar{K}}_{n}) \oplus (K \otimes {\bar{K}}_{n}) \oplus \dots \oplus (K \otimes {\bar{K}}_{n}) .$ By Remark 1, we can obtain a $G K$ -decomposition of $K \otimes {\bar{K}}_{n} .$ Further, $[(K_{m} \otimes {\bar{K}}_{n}) \times K_{r}] \otimes {\bar{K}}_{s}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{s}$ = $(K \otimes {\bar{K}}_{s}) \oplus (K \otimes {\bar{K}}_{s}) \oplus \dots \oplus (K \otimes {\bar{K}}_{s}) .$ A $G K$ -decomposition of $K \otimes {\bar{K}}_{s}$ follows from Remark 1. Thus, all together gives a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) .$

Lemma 6. A $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ exists, if $m \equiv 2, 3 (\mod 4),$ $m \neq 2,$ $n, s \equiv 1 (\mod 2)$ and $r \equiv 0, 1 (\mod 8) .$

Proof. A $G K$ -decomposition of $K_{r} \otimes {\bar{K}}_{s}$ can be obtained from Theorem 1. We then write, $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ = $(K_{m} \otimes {\bar{K}}_{n}) \times (K \oplus K \oplus \dots \oplus K)$ = $[(K_{m} \otimes {\bar{K}}_{n}) \times K] \oplus [(K_{m} \otimes {\bar{K}}_{n}) \times K] \oplus \dots \oplus [(K_{m} \otimes {\bar{K}}_{n}) \times K] .$ Now, we write as $[(K_{m} \otimes {\bar{K}}_{n}) \times K] ≅ K \times (K_{m} \otimes {\bar{K}}_{n}) ≅ (K \times K_{m}) \otimes {\bar{K}}_{n} .$ A kite decomposition of $K \times K_{m}$ has been derived from Lemma 1. Then $(K \times K_{m}) \otimes {\bar{K}}_{n}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{n}$ = $(K \otimes {\bar{K}}_{n}) \oplus (K \otimes {\bar{K}}_{n}) \oplus \dots \oplus (K \otimes {\bar{K}}_{n}) .$ A $G K$ -decomposition of $K \otimes {\bar{K}}_{n}$ exists according to Remark 1. Consequently, a $G K$ -decomposition exists for $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ .

Lemma 7. A $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ exists, if $m \equiv 2, 3 (\mod 4),$ $m \neq 2,$ $n, s \equiv 1 (\mod 2)$ and $r \equiv 4, 5 (\mod 8) .$

Proof. We write $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) ≅ [(K_{m} \otimes {\bar{K}}_{n}) \times K_{r}] \otimes {\bar{K}}_{s} .$ Now $(K_{m} \otimes {\bar{K}}_{n}) \times K_{r} ≅ K_{r} \times (K_{m} \otimes {\bar{K}}_{n}) ≅ (K_{r} \times K_{m}) \otimes {\bar{K}}_{n} .$ By Theorem 2, we have a $G K$ -decomposition of $K_{r} \times K_{m} .$ Then $(K_{r} \times K_{m}) \otimes {\bar{K}}_{n}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{n}$ = $(K \otimes {\bar{K}}_{n}) \oplus (K \otimes {\bar{K}}_{n}) \oplus \dots \oplus (K \otimes {\bar{K}}_{n}) .$ A $G K$ -decomposition of $K \otimes {\bar{K}}_{n}$ can be obtained from Remark 1. Further, $[(K_{m} \otimes {\bar{K}}_{n}) \times K_{r}] \otimes {\bar{K}}_{s}$ = $(K \oplus K \oplus \dots \oplus K) \otimes {\bar{K}}_{s}$ = $(K \otimes {\bar{K}}_{s}) \oplus (K \otimes {\bar{K}}_{s}) \oplus \dots \oplus (K \otimes {\bar{K}}_{s})$ . A $G K$ -decomposition of $K \otimes {\bar{K}}_{s}$ follows from Remark 1. Thus the above describes a $G K$ -decomposition for $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) .$

4. Main Result

Theorem 3. There exists a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ if and only if $n^{2} s^{2} m (m - 1) r (r - 1) \equiv 0 (\mod 8) .$

Proof. Necessity: The number of edges of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ is $\frac{n^{2} s^{2} m (m - 1) r (r - 1)}{2}$ and number of edges of a kite $K$ is $4$ . Therefore, the edge divisibility condition for a graph $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$ is $\frac{n^{2} s^{2} m (m - 1) r (r - 1)}{8}$ .

Sufficiency: It has been derived from Lemmas 2 – 7.

5. Conclusion

In Section 3, we give a complete solution for the existence of a $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s}) .$ In future, one can find the existence of a $G K$ -factorization of Tensor product complete multipartite graph.

Funding Information

No funds, grants, or other support were received during preparation of this manuscript.

Conflict of Interest

The author have no conflict of interest.

References:

Bermond, J.C. and Schönheim, J., 1977. G-decomposition of Kn, where G has four vertices or less. Discrete Mathematics, 19(2), pp.113-120.
Wang, H. and Chang, Y., 2006. Kite-group Divisible Designs of Type g t u 1. Graphs & Combinatorics, 22(4), 545-571.
Wang, H. and Chang, Y., 2008. (K-3+ e, lambda)-group divisible designs of type g (t) u (1). Ars Combinatoria, 89, pp.63-88.
Wang, L., 2010. On the Existence of Resolvable (K+ e)-Group Divisible Designs. Graphs & Combinatorics, 26(6), pp.879-889.
Fu, C.M., Hsu, Y.F., Lo, S.W. and Huang, W.C., 2013. Some gregarious kite decompositions of complete equipartite graphs. Discrete Mathematics, 313(5), pp.726-732.
Gionfriddo, M. and Milici, S., 2013. On the existence of uniformly resolvable decompositions of Kv and Kv- I into paths and kites. Discrete Mathematics, 313(23), pp.2830-2834.
Colbourn, C.J., Ling, A. C. and Quattrocchi, G., 2005. Embedding path designs into kite systems. Discrete Mathematics, 297(1-3), pp.38-48.
Gionfriddo, L. and Lindner, C. C., 2005. Nesting kite and $4$ -cycle systems. The Australasian Journal of Combinatorics, 33, pp.247-254.
Kucukcifci, S. and Lindner, C.C., 2002. The Metamorphosis of Lambda-fold Block Designs with Block Size Four into Lambda-fold Kite Systems. Journal of Combinatorial Mathematics and Combinatorial Computing, 40, pp.241-252.
Faro, G.L. and Tripodi, A., 2006. The Doyen–Wilson theorem for kite systems. Discrete mathematics, 306(21), pp.2695-2701.
G. Ragusa, G., 2010. Complete simultaneous metamorphosis of $λ$ -fold kite systems. Journal of Combinatorial Mathematics and Combinatorial Computing, 73, pp.159-180.
Elakkiya, A.T. and Muthusamy, A., 2016. Gregarious kite decomposition of tensor product of complete graphs. Electronic Notes in Discrete Mathematics, 53, pp.83-96.
Elakkiya, A. T. and Muthusamy, A., 2020. $G K$ -factorization of tensor product of complete graphs. Discussiones Mathematicae Graph Theory, 40, pp.7-24.

[1] Bermond, J.C. and Schönheim, J., 1977. G-decomposition of Kn, where G has four vertices or less. Discrete Mathematics, 19(2), pp.113-120.

[2] Wang, H. and Chang, Y., 2006. Kite-group Divisible Designs of Type g t u 1. Graphs & Combinatorics, 22(4), 545-571.

[3] Wang, H. and Chang, Y., 2008. (K-3+ e, lambda)-group divisible designs of type g (t) u (1). Ars Combinatoria, 89, pp.63-88.

[4] Wang, L., 2010. On the Existence of Resolvable (K+ e)-Group Divisible Designs. Graphs & Combinatorics, 26(6), pp.879-889.

[5] Fu, C.M., Hsu, Y.F., Lo, S.W. and Huang, W.C., 2013. Some gregarious kite decompositions of complete equipartite graphs. Discrete Mathematics, 313(5), pp.726-732.

[6] Gionfriddo, M. and Milici, S., 2013. On the existence of uniformly resolvable decompositions of Kv and Kv- I into paths and kites. Discrete Mathematics, 313(23), pp.2830-2834.

[7] Colbourn, C.J., Ling, A. C. and Quattrocchi, G., 2005. Embedding path designs into kite systems. Discrete Mathematics, 297(1-3), pp.38-48.

[8] Gionfriddo, L. and Lindner, C. C., 2005. Nesting kite and $4$ -cycle systems. The Australasian Journal of Combinatorics, 33, pp.247-254.

[9] Kucukcifci, S. and Lindner, C.C., 2002. The Metamorphosis of Lambda-fold Block Designs with Block Size Four into Lambda-fold Kite Systems. Journal of Combinatorial Mathematics and Combinatorial Computing, 40, pp.241-252.

[10] Faro, G.L. and Tripodi, A., 2006. The Doyen–Wilson theorem for kite systems. Discrete mathematics, 306(21), pp.2695-2701.

[11] G. Ragusa, G., 2010. Complete simultaneous metamorphosis of $λ$ -fold kite systems. Journal of Combinatorial Mathematics and Combinatorial Computing, 73, pp.159-180.

[12] Elakkiya, A.T. and Muthusamy, A., 2016. Gregarious kite decomposition of tensor product of complete graphs. Electronic Notes in Discrete Mathematics, 53, pp.83-96.

[13] Elakkiya, A. T. and Muthusamy, A., 2020. $G K$ -factorization of tensor product of complete graphs. Discussiones Mathematicae Graph Theory, 40, pp.7-24.

Contents

Journal of Combinatorial Mathematics and Combinatorial Computing

A Note on Decomposition of Tensor Product of Complete Multipartite Graphs into Gregarious Kite

Abstract

1. Introduction

2. Preliminary Results

3. Existence of $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$

4. Main Result

5. Conclusion

Funding Information

Conflict of Interest

References:

Information

Guidelines

CP Initiatives

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Contents

Journal of Combinatorial Mathematics and Combinatorial Computing

A Note on Decomposition of Tensor Product of Complete Multipartite Graphs into Gregarious Kite

Abstract

1. Introduction

2. Preliminary Results

3. Existence of GK-decomposition of (Km⊗K¯n)×(Kr⊗K¯s)

4. Main Result

5. Conclusion

Funding Information

Conflict of Interest

References:

Information

Guidelines

CP Initiatives

Follow CP

3. Existence of $G K$ -decomposition of $(K_{m} \otimes {\bar{K}}_{n}) \times (K_{r} \otimes {\bar{K}}_{s})$