Graph Decomposition and graph factor

If a graph G is H-decomposable does it imply that G has H-factor?

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1 Answer

No. An $H$-factor is a set of vertex-disjoint copies of $H$ that partition the vertex set of a graph, while an $H$-decomposition is a set of edge-disjoint copies of $H$ that partition the edge set of a graph.

For example, let $H = K_3$ and let $G$ be the $5$-vertex graph consisting of two triangles with a point in common. Then $G$ clearly has an $H$-decomposition, but, since $3 \nmid 5$, it cannot have an $H$-factor.

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Say i have a graph G with 3m vertices, and that G is $C_3$-decomposable. Can i say now that G has \$C_3_-factor? – kim_kibun Feb 21 at 14:44
@kim_kibun Again, not necessarily.  Let $G$ be the $6$-vertex graph obtained by taking a triangle and adding three vertices, each adjacent to two of the vertices of the triangle.  (More formally, let the vertices of the triangle be $u$, $v$, and $w$, and let the other three vertices be $a$, $b$, and $c$.  Let $a$ be adjacent to $u$ and $v$, $b$ to $v$ and $w$, and $c$ to $w$ and $u$.)  Then $G$ has a $K_3$-decomposition but no $K_3$-factor. – Andrew Uzzell Feb 21 at 18:13
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