### C_4 decomposition of the tensor product of complete graphs

#### Abstract

Let *G* be a simple and finite graph. A graph is said to be *decomposed* into subgraphs *H*_{1} and *H*_{2} which is denoted by *G* = *H*_{1} ⊕ *H*_{2}, if *G* is the edge disjoint union of *H*_{1} and *H*_{2}. If *G* = *H*_{1} ⊕ *H*_{2} ⊕ *H*_{3} ⊕ ... ⊕ *H _{k}*, where

*H*

_{1},

*H*

_{2},

*H*

_{3}, ...,

*H*are all isomorphic to

_{k}*H*, then

*G*is said to be

*H*-decomposable. Futhermore, if

*H*is a cycle of length

*m*then we say that

*G*is

*C*-decomposable and this can be written as

_{m}*C*|

_{m}*G*. Where

*G*×

*H*denotes the tensor product of graphs

*G*and

*H*, in this paper, we prove the necessary and sufficient conditions for the existence of

*C*

_{4}-decomposition of

*K*×

_{m}*K*. Using these conditions it can be shown that every even regular complete multipartite graph

_{n}*G*is

*C*

_{4}-decomposable if the number of edges of

*G*is divisible by 4.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.1.2

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