[10010CS 531200] Graph Theory: Autumn 2011
7.2 Homework Assignment
1. (7.2.3) For n > 1, prove that Kn,n has (n 1)!n!/2 Hamiltonian cycles.
2. (7.2.5) Prove that every 5-vertex path in the dodecahedron lies in a Hamiltonian
cycle.
3. (7.2.12) Determine wh
[10010CS 531200] Graph Theory: Autumn 2011
6.3 Homework Assignment
1. (6.3.2) A graph G is k -degenerate if every subgraph of G has a vertex of degree at
most k . Prove that every k degenerate graph is k + 1-colorable.
2. (6.3.4) Determine the crossing nu
[10010CS 531200] Graph Theory: Autumn 2011
3.1 Additional Homework Assignment
1. (3.1.5) Prove that (G)
n(G)
(G)+1
for every graph G.
2. (3.1.9) Prove that every maximal matching in a graph G has at least (G)/2 edges.
[10010CS 531200] Graph Theory: Autumn 2011
5.3 Homework Answer Correction
1. (5.3.4)
a) Prove that (Cn ; k ) = (k 1)n + (1)n (k 1).
b) For H = G K1 , prove that (H ; k ) = k(G; k 1). From this and part (a),
nd the chromatic polynomial of the wheel Cn K1 .
[10010CS 531200] Graph Theory: Autumn 2011
3.2 Homework Assignment
1. (3.2.1) Using nonnegative edge weights, construct a 4-vertex weighted graph in which
the matching of maximum weight is not a matching of maximum size.
2. (3.2.2) Show how to use the Hun
Four Color Theorem
Every planar graph is 4-colorable.
Proof of Four Color Theorem
1. Since adding edges does not make ordinary coloring
easier, to prove the Four Color Theorem it suffices
to prove that all triangulations are 4-colorable.
2. A triangulatio
Theorem 6.3.1
Every planar graph is 5-colorable.
Proof. 1. We use induction on n(G), the number of
nodes in G.
2. Basis Step: All graphs with n(G) 5 are 5-colorable.
3. Induction Step: n(G) > 5.
4. G has a vertex, v, of degree at most 5
because
(Theorem 6
[10010CS 531200] Graph Theory: Autumn 2011
5.3 Homework Assignment
1. (5.3.1) Compute the chromatic polynomials of the graphs below.
2. (5.3.3) Prove that k 4 4k 3 + 3k 2 is not a chromatic polynomial.
3. (5.3.4)
a) Prove that (Cn ; k ) = (k 1)n + (1)n (k
[10010CS 531200] Graph Theory: Autumn 2011
7.1 Homework Assignment
1. (7.1.1) For each graph G below, compute (G).
2. (7.1.18) Give an explicit edge-coloring to prove that (Kr,s ) = (Kr,s ).
3. (7.1.19) Prove that for every simple bipartite graph G, there
[CS 5312] Graph Theory: Autumn 2010
1st exam (close book)
Examination Date: Oct. 25, 2010
Time: 15:20-17:10
1. For each k > 1, construct a k -regular simple graph having no 1-factor. (15%)
2. Let the matrix shown below be the input of the Hungarian Algori
[10010CS 531200] Graph Theory: Autumn 2011
6.1 Homework Assignment
1. (6.1.1) Prove or disprove:
a) Every subgraph of a planar graph is planar.
b) Every subgraph of a nonplanar graph is nonplanar.
2. (6.1.5) Prove or disprove: A plane graph has a cut-vert
[10010CS 531200] Graph Theory: Autumn 2011
5.2 Homework Assignment
1. (5.2.2) Prove that a simple graph is a complete k -partite graph (k 2) if and only if
it has no 3-vertex induced subgraph with one edge.
2. (5.2.5) Find a subdivision of K4 in the Grtzs
[100CS 531200] Graph Theory: Autumn 2011
1st exam (close book)
Examination Date: Oct. 31, 2011
Time: 09:30-12:00
1. Please answer the following questions.
1) Find two errors in the proof of Theorem 1, and briey explain your reasons. (10%)
2) Why is the ar
[10010CS 531200] Graph Theory: Autumn 2011
4.1 Homework Assignment
1. (4.1.2) Give a counterexample to the following statement, add a hypothesis to correct
it, and prove the corrected statement: If e is a cut-edge of G, then at least one
endpoint of e is
[10010CS 531200] Graph Theory: Autumn 2011
4.2 Homework Assignment
1. (4.2.4) Prove or disprove: If P is a u, v -path in a 2-connected graph G, then there is
a u, v -path Q that is internally disjoint from P .
2. (4.2.12) Use Mengers Theorem to prove that
[10010CS 531200] Graph Theory: Autumn 2011
5.1 Homework Assignment
1. (5.1.2) Prove that the chromatic number of a graph equals the maximum of the
chromatic numbers of its components.
2. (5.1.7) Construct a graph G that is neither a complete graph nor an
[10010CS 531200] Graph Theory: Autumn 2011
6.2 Homework Assignment
1. (6.2.4) For each graph below, prove nonplanarity or provide a convex embedding.
2. (6.2.5) Determine the minimum number of edges that must be deleted from the Petersen graph to obtain a
[CS 5312] Graph Theory: Autumn 2010
Final Exam (close book)
Examination Date: Jan. 10, 2011
Time: 15:20-17:10
1. Please answer the following questions.
1) Why is the argument 8 in the proof of Theorem 1 correct? (6%)
2) Why do we need to put y at a point
[CS 5312] Graph Theory: Autumn 2010
1st exam (close book)
Examination Date: Nov. 29, 2010
Time: 15:20-17:10
1. Please answer the following questions. (60%)
1) Why is the argument 2 in the proof of Theorem 1 correct?
2) Why is the argument 4 in the proof o
Matchings
Matching: A matching in a graph G is a set of
non-loop edges with no shared endpoints
Maximal & Maximum Matchings
Maximal Matching: A maximal matching in a graph
is a matching that cannot enlarged by adding an edge
Maximum Matching: A maximum
Mycielskis Construction
Mycielskis Construction: From a simple graph G,
Mycielskis Construction produces a simple graph G
containing G. Beginning with G having vertex set
cfw_v1, v2, ,vn, add vertices U=cfw_u1, u2, ,un and one
more vertex w. Add edges to
[10010CS 531200] Graph Theory: Autumn 2011
3.3 Homework Assignment
1. (3.3.2) Exhibit a maximum matching in the graph below, and use a result in this
section to give a short proof that it has no larger matching.
2. (3.3.7) For each k > 1, construct a k -r