Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative combinatorics
MATH 18.315

Spring 2009
HOMEWORK 2 (18.315, FALL 2005)
1) Let X R2 be the set of n 4 points in general positions. Supposed points in X are
colored with two colors, such that for every four points there is a line separating points of
dierent color. Prove that there exist a line s
Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative combinatorics
MATH 18.315

Spring 2009
HOMEWORK 3 (18.315, FALL 2005)
1) Decide whether a rectangle [50 60] can be tiles with rectangles
b) [5 8]
a) [20 15]
d) [2 2 2 + 2]
c) [6.25 15]
e) Find and prove a general criterion for tileability of a rectangle [a b] with rectangular tiles
[p q ].
2)
Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative combinatorics
MATH 18.315

Spring 2009
HOMEWORK 1
(18.315, FALL 2005)
Def. A proper coloring of a graph is a coloring of vertices with no monochromatic edges.
A grid graph Gm,n is a product of a mpath and a npath.
1) Let c(n) be the number of proper colorings of Gn,n with 3 colors. Prove
2
a)