08/24/15
Math 412
HW1
Due Wednesday, September 9, 2015
Solve four of the next ve problems.
1. Determine which pairs of graphs below are isomorphic.
(a)
(b)
(c)
2. Consider the following four families of simple graphs: A = cfw_complete graphs, B =
cfw_path
10/14/15
Math 412
HW7
Due Wednesday, October 21, 2015
Solve four of the next ve problems.
1. Let G be an n-vertex 3-regular graph with at most 5 cut-edges. Prove that G has a
matching with at least 0.5n 1 edges. What about 6 cut-edges?
2. Prove that for e
09/30/15
Math 412
HW5
Due Wednesday, October 07, 2015
Solve four of the next ve problems.
1. Use Prims Algorithm and Kruskals Algorithm to nd minimum spanning trees in the
weighted graph on the other side of this sheet (show the sequence in which edges ar
09/23/15
Math 412
HW4
Due Wednesday, September 30, 2015
Solve four of the next ve problems.
1. Let n 3 and G be an n-vertex graph. Prove that the following are equivalent.
(A) G is connected and has exactly one cycle.
(B) G is connected and has n edges.
(
10/07/15
Math 412
HW6
Due Wednesday, October 14, 2015
Solve four of the next ve problems.
1. Let A be a 0, 1-matrix. A line in A is a row or a column. Two 1s in A are independent
if no line contains both of them. Prove that the maximum number of pairwise
For a digraph G = (V, E) and v V , denote by E + (v) the set of edges leaving v
and by E (v) the set of edges entering v.
A network G = cfw_V, E, s, t, b = cfw_b(e)eE is a directed graph (V, E) with a source
vertex s, a sink vertex t, and non-negative ca
09/16/15
Math 412
HW3
Due Wednesday, September 23, 2015
Solve four of the next ve problems.
1. Given a nonincreasing list d = (d1 , . . . , dn ) of nonnegative integers and 1 k n, let
d(k) be obtained from d by deleting dk and subtracting 1 from dk larges
10/28/15
Math 412
HW9
Due Wednesday, November 04, 2015
Solve four of the next ve problems.
1. (a) Prove that every 3-connected simple graph G has a cycle C such that G V (C)
is connected. (Hint: Choose a cycle C with a largest component of G V (C).
(b) Fo
09/09/15
Math 412
HW2
Due Wednesday, September 16, 2015
Solve four of the next ve problems.
1. Prove that every connected graph G has a walk that traverses each edge of G exactly
four times. Give an example of a connected graph H that does not have a walk
Theorem 1 (Mantel, 1907). Let f (n) be the maximum number of edges in a simple n-vertex
2
graph with no triangles. Then f (n) = n .
4
Proof. The fact that f (n)
n2
4
follows from the example of the complete bipartite
graph K n/2 , n/2 , since it is bipar
Course Outline FALL 2015
MATH 412
INTRODUCTION TO GRAPH THEORY
Sections X13 and X14: 12 Noon MWF, 341 Altgeld Hall.
Instructor: A. Kostochka, 234 Illini Hall, 265-8037, kostochk@math.uiuc.edu.
Oce hours: tentatively MWF 3:104:10 and by appoinment.
Web pag
11/04/15
Math 412
HW10
Due Wednesday, November 18, 2015
Solve four of the next ve problems.
1. Using maximum ows (solution without ows does not count!), nd a maximum
matching in the bipartite graph below. Prove that the matching is optimal. Find a smalles
10/21/15
Math 412
HW8
Due Wednesday, October 28, 2015
Solve four of the next ve problems.
1. Prove that a simple graph G with at least four vertices is 3-connected if and only if
for every triple (x, y, z) of distinct vertices and any edge e not incident
Math 412
HW7
Due Friday, March 25, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. let G be a bipartite graph. Prove that if (G) n(G)/4, then 0 (G)
Math 412
HW1
Due Friday, January 29, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. Determine which pairs of graphs below are isomorphic.
(a)
(b)
(
10/21/15
Math 412
HW8
Due Wednesday, October 28, 2015
Solve four of the next five problems.
1. Prove that a simple graph G with at least four vertices is 3-connected if and only if
for every triple (x, y, z) of distinct vertices and any edge e not inciden
09/23/15
Math 412
HW4
Due Wednesday, September 30, 2015
Solve four of the next five problems.
1. Let n 3 and G be an n-vertex graph. Prove that the following are equivalent.
(A) G is connected and has exactly one cycle.
(B) G is connected and has n edges.
09/30/15
Math 412
HW5
Due Wednesday, October 07, 2015
Solve four of the next five problems.
1. Use Prims Algorithm and Kruskals Algorithm to find minimum spanning trees in the
weighted graph on the other side of this sheet (show the sequence in which edge
10/28/15
Math 412
HW9
Due Wednesday, November 04, 2015
Solve four of the next five problems.
1. (a) Prove that every 3-connected simple graph G has a cycle C such that G V (C)
is connected. (Hint: Choose a cycle C with a largest component of G V (C).
(b)
08/24/15
Math 412
HW1
Due Wednesday, September 9, 2015
Solve four of the next five problems.
1. Determine which pairs of graphs below are isomorphic.
(a)
(b)
(c)
2. Consider the following four families of simple graphs: A = cfw_complete graphs, B =
cfw_pa
09/16/15
Math 412
HW3
Due Wednesday, September 23, 2015
Solve four of the next five problems.
1. Given a nonincreasing list d = (d1 , . . . , dn ) of nonnegative integers and 1 k n, let
d(k) be obtained from d by deleting dk and subtracting 1 from dk larg
10/14/15
Math 412
HW7
Due Wednesday, October 21, 2015
Solve four of the next five problems.
1. Let G be an n-vertex 3-regular graph with at most 5 cut-edges. Prove that G has a
matching with at least 0.5n 1 edges. What about 6 cut-edges?
2. Prove that for
Math 412
HW2
Due Friday, February 5, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. Let P and Q be two maximum length paths in a connected graph G.
Math 412
HW3
Due Friday, February 19, 2016
Students in the three credit hour course must solve five of the six problems. Students in the four
credit hour course must solve all six problems.
1. For k 2, prove that every k-regular bipartite graph has no cut
11/04/15
Math 412
HW10
Due Wednesday, November 18, 2015
Solve four of the next five problems.
1. Using maximum flows (solution without flows does not count!), find a maximum
matching in the bipartite graph below. Prove that the matching is optimal. Find a