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
10/21/15
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
Graph Theory lecture notes
1
11
Definitions and examples
Definitions
Definition 1.1. A graph is a set of points, called vertices, together with a collection of lines,
called edges, connecting some of the points. The set of vertices must not be empty.
If G
Course Outline SPRING 2017
MATH 412
INTRODUCTION TO GRAPH THEORY
Section C13: 10 am MWF, 245 Altgeld Hall.
Instructor: A. Kostochka, 234 Illini Hall, 265-8037, [email protected]
Office hours: tentatively MWF 34pm and by appoinment.
Web page: http:/w
Theorem 1 (Shannon) Let G = (V, E) be a (multi)graph with no loops, and the maximum
degree of G be . Then the edges of G can be properly colored with at most 3/2 colors.
PROOF. If 1, then the statement is evident. Let 2. We will prove the statement
by ind
03/29/17
Math 412
HW8
Due Wednesday, April 12, 2017
Solve four of the next five problems.
1. Let G be the network with the flow drawn on the left of Fig. 1. Write the flow as a linear
combination of flows along cycles, s, t-paths and t, s-paths.
2. In the
03/08/17
Math 412
HW7
Due Wednesday, March 29, 2015
Solve four of the next five problems.
1. Prove that for every simple graph G with maximum degree at most 3, 0 (G) = (G).
2. Let G be a simple bipartite graph. Prove that if (G) n(G)/3, then (G) = (G).
3.
01/25/17
Math 412
HW2
Due Wednesday, February 01, 2017
Solve four of the next five problems.
1. Prove or disprove:
(a) Every connected graph G has a closed walk that traverses each edge of G exactly four
times;
(b) Every connected graph G has a closed wal
02/15/17
Math 412
HW5
Due Wednesday, March 01, 2017
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 edges
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.
(
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
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, [email protected]
Oce hours: tentatively MWF 3:104:10 and by appoinment.
Web pag
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
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
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
09/16/15
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
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
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
02/08/17
Math 412
HW4
Due Wednesday, February 15, 2017
Solve four of the next five problems.
1. Using a generalized de Bruijn graph with alphabet cfw_0, 1, 2 instead of cfw_0, 1,
find a cyclic arrangement of 27 digits 0, 1, and 2 such that all 27 strings
03/29/17
Math 412
HW8
Due Wednesday, April 12, 2017
Solve four of the next five problems.
1. Let G be the network with the flow drawn on the left of Fig. 1. Write the flow as a linear
combination of flows along cycles, s, t-paths and t, s-paths.
2. In the
04/12/17
Math 412
HW 9
Due Wednesday, April 26, 2017
Solve four of the next five problems.
1. Determine whether the following graphs are planar: (a) The graph obtained from the
complete bipartite graph K4,4 by deleting a perfect matching; (b) The graph ob
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
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
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
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)
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
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.