Homework 1 Solutions
1. Prove or disprove: If every vertex of G has degree 2, then G is a cycle.
Solution: This is false. A graph with two components each of which is a cycle is a counterexample.
2. Prove that a bipartite graph has a unique bipartition (u
Homework 7 Solutions
Problem 1. Find a graph on 100 vertices with 98 cut vertices.
Solution: P100 .
Problem 2. Find a formula for the number of spanning trees of a graph in terms of the
number of spanning trees of each block.
Solution: Let G be a graph an
Homework 9 Solutions
Problem 1. Find a strongly connected directed graph with 100 vertices and the fewest
possible edges.
Solution: Let D be a directed cycle on 100 vertices. If D is any strongly connected digraph
with |E(D )| = 100, then every vertex in
Homework 10 Solutions
Problem 1. In the following digraph, each edge is labelled as (e) | c(e) where is a
(s, t)-ow and c(e) is the capacity of the edge e. Either prove that there is an augmenting
path from s to t of nd a cut with capacity equal to the ex
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Connectivity
2-connectivity
Separation: A separation of G of order k is a pair of subgraphs (H, K) with H K = G
and E(H K) = and |V (H) V (K)| = k. Such a separation is proper if V (H) \ V (K)
and V (K) \ V (H) are nonempty.
Observation 4.1 G has a prop
Homework 8 Solutions
Problem 1. Find a 100-connected bipartite graph G for which |V (G)| is minimum.
Solution: The graph K100,100 is the smallest 100-connected bipartite graph. Every 100connected graph has minimum degree 100, so every 100-connected bipart
Homework 5 Solutions
Problem 1. If G is a graph with a maximum matching of size 2k, what is the smallest
possible size of a maximal matching in G?
Solution: The answer is k. To construct such a graph, take a graph with k components, each
of which is a thr
Homework 2 Solutions
1. Determine the maximum size of an independent set in Petersen.
Solution: Petersen is a 3-regular graph on 15 vertices. Were it to contain an independent
set X of size 5, then every edge of the graph must be incident with X, so then
Homework 3
Problem 1. Let G be a graph with the property that every subgraph of G has a vertex of
degree 1. Show that G is a forest.
Solution: If G contains a cycle C G, then C has all vertices of degree 2, giving us a
contradiction. Thus, G is a forest.
Homework 4 Solutions
Problem 1. In the weighted graph from the gure below, nd the sequence of edge weights
selected when both Kruskals algorithm is run, and when Dijkstras algorithm is run.
3
9
2
8
5
7
10
11
6
1
4
r
13
14
12
Solution: Kruskals algorithm c
Homework 6 Solutions
1. Exhibit a Marriage System which has more than one stable marriage.
Solution: Let (cfw_a, b, cfw_x, y) be a bipartition of K2,2 and dene a system of preferences as
follows:
a : x>y
b : y>x
x : b>a
y : a>b
Both perfect matchings of t
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Graph Basics
What is a graph?
Graph: a graph G consists of a (nite) set of vertices, denoted V (G), a (nite) set of edges,
denoted E(G), and a relation called incidence so that each edge is incident with either one or
two vertices, called its ends. For
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Directed Graphs
What is a directed graph?
Directed Graph: A directed graph, or digraph, D, consists of a set of vertices V (D), a set
of edges E(D), and a function which assigns each edge e an ordered pair of vertices (u, v).
We call u the tail of e, v
2
Trees
What is a tree?
Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest.
Theorem 2.1 If G is a forest, then comp(G) = |V (G)| |E(G)|.
Proof: We proceed by induction on |E(G)|. As a base, if |E(G)| = 0, then every compon
The Stable Marriage Theorem
System of Preferences: If G is a graph, a system of preferences for G is a family cfw_>v vV (G)
so that each >v is a linear ordering of N (v). If u, u N (v) and u >v u , we say that v
prefers u to u .
Marriage Systems and Stabl
Advice for solving graph theory problems
Proving theorems from scratch is a dicult - but rewarding - art. It requires focus, patience,
and inspiration. With a hard problem, it is impossible to simply read out the question and
then start writing the soluti
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Graph Basics
What is a graph?
Graph: a graph G consists of a set of vertices, denoted V (G), a set of edges, denoted E(G),
and a relation called incidence so that each edge is incident with either one or two vertices
- its ends. We assume that V (G) and
Mathematical Induction
Induction is an incredibly powerful tool for proving theorems in discrete mathematics. In
this document we will establish the proper framework for proving theorems by induction,
and (hopefully) dispel a common misconception.
Basic i
3
Matchings
Halls Theorem
Matching: A matching in G is a subset M E(G) so that no edge in M is a loop, and no
two edges in M are incident with a common vertex. A matching M is maximal if there is
no matching M with M M and maximum if there is no matching
Midterm Solutions
Math 345, Graph Theory I
Instructor: Matt DeVos
Name (print):
Signature:
Problem Score Value
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b
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1. (3 points) Find a maximum matching in the above graph.
a, d, g, j, m