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$$(G_m,H_m)$$-multifactorization of $$\lambda K_m$$. (English) Zbl 1195.05061
Summary: A $$(G,H)$$-multifactorization of $$\lambda K_m$$ is a partition of the edge set of $$\lambda K_m$$ into $$G$$-factors and $$H$$-factors with at least one $$G$$-factor and one $$H$$-factor. Atif Abueida and Theresa O’. Neil [“Multidecomposition of $$\lambda K_m$$ into small cycles and claws”, Bull. Inst. Comb. Appl. 49, 32–40 (2007; Zbl 1112.05084)] have conjectured that for any integer $$n\geq 3$$ and $$m\geq n$$, there is a $$(G_n,H_n)$$-multidecomposition of $$\lambda K_m$$ where $$G_n= K_{1,n-1}$$ and $$H_n= C_n$$. In this paper it is shown that the above conjecture is true for $$m=n$$ when
(i)
$$G_m=K_{1,m-1}$$; $$H_m=K_m$$,
(ii)
$$G_m= H_{1,m-1}$$; $$H_m= P_m$$ and
(iii)
$$G_m= P_m$$; $$H_m= C_m$$.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
multifactorization; multidecomposition; stars; paths; cycles