Math 5707 : Graph Theory , Spring 2017 Midterm 2

Math 5707 : Graph Theory , Spring 2017 Midterm 3

Math 5707 : Graph Theory , Spring 2017

1.1 Problem Fix a loopless multidigraphD = (V,A, φ). Let f : V → N be a configuration. Let h = ∑ f . Let n = |V |. Assume that n > 0. Let ` = (`1, `2, . . . , `k) be a legal sequence for f having length k ≥ ( n+ h− 1 n− 1 ) . Prove the following: (a) There exist legal sequences (for f) of arbitrary length. (b) Let q be a vertex of D such that for each vertex u ∈ V , there exists a path from u t...

Math 5707 : Graph Theory , Spring 2017 Homework 5

1.1 Problem Fix a loopless multidigraphD = (V,A, φ). Let f : V → N be a configuration. Let h = ∑ f . Let n = |V |. Assume that n > 0. Let ` = (`1, `2, . . . , `k) be a legal sequence for f having length k ≥ ( n+ h− 1 n− 1 ) . Prove the following: (a) There exist legal sequences (for f) of arbitrary length. (b) Let q be a vertex of D such that for each vertex u ∈ V , there exists a path from u t...

UMN, Spring 2017, Math 5707: Lecture 16 (flows and cuts in networks)

1. Network flows 1 1.

MATH 5707 Midterm 2

Proof. Let n = |V |, and assume without loss of generality that |E| = n (else we could remove edges from G until |E| = n and apply the argument). Also assume WLOG that V ∩ E = ∅ (else, we can rename the vertices). Note that we must have n ≥ 3, since a simple graph with 2 or fewer vertices can have at most 1 edge. Now we seek a bijection f : V → E with each v ∈ V satisfying v / ∈ f(v). Construct...