18.314     PROBLEM SET 5    (due Thursday November 17, 2005)



Problem 1. [Bona, p 200, (25)]  Prove that in any simple graph, there
are two vertices with the same degree.  (A simple graph is a graph that 
has no loops and has no multiple edges.)


Problem 2. [Bona, p. 229, (23)]  Prove that a tree has more leaves
than vertices of degree at least three.


Problem 3.  Let G be a connected graph with positive weights on edges.
We would like to find a subgraph H of G such that (1) the graph
G - H obtained from G by removing edges of H is connected;
(2) the sum of weights of edges in H is as big as possible for
subgraphs that satisfy (1).
     Let us try to use the greedy algorithm:  Pick an edge e_1 of G of
maximal weight such that G - {e_1} is connected; then pick an edge e_2 
of maximal weight such that G - {e_1,e_2} is still connected, etc.
Will this algorithm always produce a subgraph with the desired property?
Prove that the greedy algorithm always works or find a counterexample.


Problem 4.  Find a formula for the number of labeled trees on 2n vertices
such that each vertex has either degree 3 or degree 1 (leaf). 
For example, for n=2, there are 4 such trees:  They have 1 central
node (labeled by a number 1, 2, 3, or 4) connected with 3 leaves.
(Hint:  Try to count Prufer codes of such trees.)


Problem 5. Prove that the number of spanning trees in the complete 
bipartite graph K_mn is m^{n-1} n^{m-1}.  (Hint: Use the Matrix-Tree 
theorem or try to modify Prufer coding for bipartite graphs.)


Problem 6 (bonus)  Let PF_n be the set of sequences (a_1,...,a_n)
of positive integers such that the increasing rearrangement
c_1 <= ... <= c_n of a_1,...,a_n satisfies c_i <= i, for i=1,...,n.  
For example, the set PF_3 consists of 16 sequences: FP_3 = {123, 132, 
213, 231, 312, 321, 113, 131, 311, 112, 121, 211, 122, 212, 221, 111}.

Find a bijection between the set PF_n and labeled trees on n+1 vertices
and thus show that |PF_n| = (n+1)^{n-1}.  (A partial credit will be given 
for a non-bijective proof of the claim that |PF_n| = (n+1)^{n-1}.)