臺灣大學數學系
九十二學年度博士班入學考試題 離散數學
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(20 %) The number of steps required to solve the ``Chinese rings puzzle'' with rings
satisfies and
Prove that . Hence or otherwise find a formula for
in term of . 2. (20 %) Suppose is a family of sets satisfying the Hall's condition as follows:
for all
where . If moreover that each has at least two elements, prove that the family has at least two SDR (system of distinct representatives). 3. (20 %) Prove that if a graph is -free then either or its complement is not connected. 4.
(20 %) Let be the unit-distance graph in the plane: , and two points are adjacent if and only if their Euclidean distance is . Notice that this is an infinite graph. Prove that . 5. (20 %) The (Hamiltonian) closure of an -vertex graph , denoted by , is the graph with vertex set obtained from by iteratively adding edges joining pairs of nonadjacent vertices whose degree sum is at least , until no such pair remains. Notice that the closure of is well-defined. Prove that for any , a simple -vertex graph is Hamiltonian if and only if its closure is Hamiltonian.
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calbat 2003-07-07
