| Summary: | 碩士 === 國立中央大學 === 數學系 === 84 === For integers $n \ge k \ge 1,$ we use $ C_{n,k} $ to denote the
graph with vertex set $\{ a_1, a_2, \ldots , a_n, b_1, b_2,
\ldots , b_n \}$ and edge set $\{ a_ib_j : 1 \le i \le n, j=
i-1, i-2,\ldots ,i-k \ (mod \ n)\}$. We call $C_{n,k}$ a crown.
We consider the edge decompositions of crowns (respectively,
complete bipartite graphs) into isomorphic copies of star
forests. For graphs $G$ and $H$, we use $G ---> H $ to denote
that the edges of $G$ can be decomposed into isomorphic copies
of $H$. Let $M_s$ denote a matching with $s$ edges. The main
results are the following: 1. Suppose $ l \le m$. Then $K_{l,m}
---> M_s$ if and only if $s \le l$ and $s|lm$. 2. Suppose
$SF_l$ is a star forest of size $l$ having no isolated vertex,
and has no more than $m$ components. Then $K_{l,m} ---> SF_l$.
3. Suppose $SF_l$ is a star forest of size $l$ having no
isolated vertex. Then $C_{n,l} ---> SF_l$. 4. Suppose $SF_n$ is
a star forest of size $n$ having no isolated vertices, and
$SF_n$ has at least two components. Then $C_{n,n-1}$ has an
$SF_n$-decomposition. We also consider the decomposition into
generalized stars, which are defined below. For positive
integers $l,m$, let $S_l(m)$ denote the graph with vertex set
$\{ a_0\} \cup \{ a_{i,j} : 1 \le i \le m,1 \le j \le l \}$ and
edge set $\{ a_0a_{1j} : 1 \le j \le l \} \cup \{ a_{i-1,j}a_{i,
j} : 2 \le i \le m,\,1 \le j \le l \}$. We call $S_l(m)$ a
generalized star. The results are the following: 1. $K_{tl,n}
---> S_l(2)$ if and only if 2 | tn and n>=2l if t=1, n > l if
t>=2. 2. Suppose $n$ is odd. Then $C_{n,tl} ---> S_l(2) iff t
is even.
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