4.2.23 Use Mengers Theorem (x, y ) = (x, y ) when xy E (G) to prove the KonigEgervary Theorem ( (G) = (G) when G is bipartite).
Proof. Let G be a X, Y bipartite graph. Construct H with V (H ) = V (G) cfw_a, b, E (H ) =
3.1.1 Find a maximum matching in each graph below. Prove that it is a maximum
matching by exhibiting an optimal solution to the dual problem (minimum vertex cover). Explain why this proves that the matching is optimal.
Answer. We saw
4.1.1 Give a proof or a counterexample for each statement below.
(a) Every graph with connectivity 4 is 2-connected.
Answer. True. 2 (G) = 4.
(b) Every 3-connected graph has connectivity 3.
Answer. False. K5 is 3-connected
Homework 1 solutions
1.1.10 If G is simple and disconnected, then G is connected.
Proof. Consider any two vertices u, v V (G). Well use the fact that G is disconnected to
. Since this will be true for any two vertices in G, this
show that u and v lie on
Homework 2 solutions
1.1.30 Let G be a simple graph with adjacency matrix A and incidence matrix M . Prove
that the degree of vi is the ith diagonal entry of A2 and M M T . What do the
entries in position (i, j ) of A2 and M M T say about G?
1.3.8 Which of the following are graphic sequences? Provide a construction of a proof
of impossibility for each.
(a) (5, 5, 4, 3, 2, 2, 2, 1): Using the iteration from Theorem 1.3.31, you get
(5, 5, 4, 3, 2, 2, 2, 1) (4, 3, 2, 1, 1, 2
2.1.13 Prove that every graph with diameter d has an independent set with at least
(1 + d)/2 vertices.
Proof. Consider a path of minimal length between two vertices u and v with d(u, v ) = d. This
path has d + 1 vertices, all with the
2.2.3 Let G be the graph below. Use the Matrix Tree Theorem to nd a matrix whose
determinant is (G).
and A =
8 4 1 3
4 6 2 0
1 2 5 2
3 0 2 5
So, for example
6 2 0
MATH 38, SPRING 2012
1. [20 points]
(a) Prove that if G is a simple graph of order n such that (G) + (G) n 1, then G is connected.
(Hint: Consider a vertex of maximum degree.)
(b) Show that this bound is sharp (i.e. there is no smaller w
SOLUTIONS TO 2.3 PROBLEMS
2.3.6 Assign integer weights to the edges of Kn . Let the weight of a cycle be the sum
of the weights of its edges. Prove that all cycles have even weight if and only if
the subgraph formed by the edges with odd weight
Quiz 3, Math 38, Spring 2012
Always (briey) explain or demonstrate why, unless told otherwise.
(1) For the graph below,
(a) give an optimal matching M ;
(b) give a maximum M -alternating path;
(c) give an optimal vertex covering;
(d) and jus