Download Exercise Set 3

MA2059/MA3060 Exercise Set 3
Assignment 2 - due March 12: Write down solutions for any 5 of
the following exercises, with the exception of those marked with ∗ , which
will be solved in class.
Graphs
1. In how many ways can you arrange the numbers 1, 2, 4, 5, 6, 8, 9, 10, 12, 15
in a row if a number always has priority over its double and over its triple?
2. Suppose n cuts are made across a pizza. Let pn denote the maximum
number of pieces which can result (this happens when no two cuts are parallel or
meet outside the pizza, and no three are concurrent). Find pn as a function of n.
3. A convex polyhedron P has only square and hexagonal faces. Three faces
meet at each vertex. Prove that P must have exactly 6 square faces. Give an
example of such a polyhedron P with 6 square faces and one hexagonal face (try
truncating an octahedron).
4. A k cube is the graph whose vertices are the ordered k-tuples of 0-s and
1-s, two vertices being joined if and only if they differ in exactly one coordinate.
Show that the k-cube has 2k vertices, k2k−1 edges and is bipartite. For which
values of k is the k cube planar?
5. If G is a simple (at most one edge between two points) bipartite graph
with E edges and V vertices, prove E ≤ V 2 /4.
6. An m-partite
graph is one whose vertex set V can be partitioned into m
F
V
so
that no edge has both ends in any one subset Vi . Let Tmk,m
subsets V = m
i
i=1
be the complete m-partite graph with |Vi | = k for all i ∈ {1, ..., m}.
a) Find the number of edges E(Tmk,m ) as a function of m and k.
b) If G is any complete m-partite graph of mk vertices, prove E(G) ≤ E(Tmk,m ),
with equality when G = Tm,n .
7. a) Show that all alcohols (Cn H2n+1 OH) have tree like molecules. (the
valencies of C, O, H are 4,2,1 respectively).
b) Show that the only saturated hydrocarbon molecules (tree like molecules with
atomic composition Cn Hm ) are the paraffins (m = 2n + 2).
8. Let S = {x1 , ..., xn } be a set of points in the plane such that the distance
between any two points is at least 1. Show that there are at most 3n pairs of points
at distance exactly 1. [Hint: define a graph were the points at distance exactly 1
are joined by an edge. Prove that the degree of each vertex is at most 6.]
9.∗ a) Let S(m, n) be the number of surjective functions from a set of m
elements to a set of n elements. Prove
S(m, n) =
n−1
X
(−1)k Ckn (n − k)m .
k=0
1
2
b) Let T (n) be the number of spanning trees of the complete graph with n vertices.
Prove
n−1
X
T (n) =
(−1)k−1 Ckn (n − k)k T (n − k).
k=1
n−2
c) Prove that t(n) = n
satisfies the recursion relation above and t(3) = T (3), t(4) =
T (4). Conclude T (n) = t(n).
10. a) There used to be 26 teams in the U.S. National Football League with
13 teams in each of two divisions. A League guideline said that each team’s 14game schedule should include exactly 11 games against teams in its own division
and 3 games against teams in the other division. Show that this guideline could
not be satisfied by all the teams.
b) Either find a graph that models the following or show that none exists: Each
of 102 students in a programme will be assigned the use of 1 of 35 computers, and
each of the 35 computers will be used by exactly 1 or 3 students.
11. In a chess tournament with 2n players, a total of n2 +1 games were played.
No pair of competitors played together more than once. Show that there is a group
of three players in which each pair played together during the tournament.
12.∗ For any tree T of vertex set V = {v1 , ..., vn }, we define the index
i(T ) := min{k ∈ {1, ..., n}; deg(vk ) = 1}.
The Pruffer code of the tree is a tuple of numbers (a1 , ..., an−2 ) with 1 ≤ ai ≤ n,
defined inductively as follows:
Let T1 := T . For k ∈ {1, ..., n − 1},
• Let ak := the index l ∈ {1, ..., n} such that {vi(Tk ) , vl } is an edge in Tk .
• Let Tk+1 be obtained by deleting {vi(Tk ) , vl } from Tk .
Prove that the Pruffer code
P : { trees with n vertices } → {1, ..., n}n−2
is a bijective function.
13.∗ a) Show P
that a sequence (d1 , ..., dn ) of positive integers is a degree sequence of a tree iff ni=1 di = 2(n − 1).
b) Prove that the number of trees with degree sequence (d1 , ..., dn ) is
(n − 2)!
Q
.
i (di − 1)!
14.∗ Let Gn denote the graph with vertices {x1 , x2 , ..., xn , y1 , y2 , ..., yn } and
edges {xi , yi } and {xi , xi+1 } and {yi , yi+1 } for all i. Prove that the number of
spanning trees of Gn is
√ n
√ n
1 √ (2 + 3) − (2 − 3) .
2 3
3
15.∗ Prove that any graph with V vertices and E edges contains at least
2
4E
(E − V4 ) triangles.
3V
16. Let d = (d1 , ..., dn ) be a non-increasing sequence of positive integers and
denote d0 = (d1 − 1, ..., dn − 1). Is it true that d is the degree sequence of a graph
iff d0 is the degree sequence of a graph?
17.∗ A convex polyhedron P has only triangle and pentagonal faces. Four
faces meet at each vertex. The polyhedron looks the same at each vertex (same
number of triangles and pentagonal faces, arranged in the same order up to rotation).
a) Find the number of triangles and pentagonal faces of P in terms of V , its number of vertices.
b) Prove that either P has 10 triangles and 2 pentagonal faces, or P has 20 triangles and 12 pentagonal faces.
c) In each of the two cases above, assume the polyhedron has a plane of symmetry
passing through some of its edges, and draw the graph that you obtain by halving
the polyhedron along such a plane of symmetry.
Incidence Matrices
18. (China 1996) Eight singers participate in an art festival where m songs
are performed. Each song is performed by 4 singers, and each pair of singers
performs together in the same number of songs. Find the smallest m for which
this is possible.
19. (China 1995) Twenty-one people took a test with 15 True/False questions. It is known that for every two people, there is at least one question that
both have answered correctly. Determine the minimum possible number of people
that could have correctly answered the question that most number of people are
correct on.
20.∗ (∼ EGMO 2013) Snow White and the Seven Dwarves are living in their
house in the forest. On each of 16 consecutive days, some of the dwarves worked in
the diamond mine while the remaining dwarves collected berries in the forest. No
dwarf performed both types of work on the same day. On any two different (not
necessarily consecutive) days, at least three dwarves each performed both types of
work. Further, on the first day, all seven dwarves worked in the diamond mine.
a)Prove that for any 3 days and any 3 dwarfs, at least 1 of the dwarfs must change
activity in at least one of the days.
b)For any three dwarfs, prove that each possible way to assign work to the 3 dwarfs
happened exactly twice in the 16 days.
c)Prove that, on one of these 16 days, all seven dwarves were collecting berries.
Anca Mustata, School of Mathematical Sciences, UCC
E-mail address: [email protected]