Chapter 2
Trees and Connectivity
Although the property of a graph G being connected depends only on whether G
contains a u v path for every pair u, v of vertices of G, there are varying degrees
of connectedness that a graph may possess. Some of the best-k
MATH 306: LECTURE NOTES 8
20. Nowhere-zero Flows
Let be an abelian group (ie so x + y = y + x for all x, y ). A nowhere-zero -ow
in a digraph G means a function : E (G) such that for every vertex v ,
(e) =
e + (v )
(e),
e (v )
and such that (e) = 0 for ea
MATH 306 HOMEWORK 1
For each of the following statements, decide if it is true or false, and either prove it or give
a counterexample.
1. Every simple graph with 2 vertices has two vertices with the same degree.
2. There is a simple graph with 8 vertices
MATH 306 HOMEWORK 4
1. Let G be a bipartite graph, with bipartition (A, B ). Show that the following are
equivalent:
(i) there is a matching M in G so that every vertex in A is an end of some member
of M
(ii) for every X A there are at least |X | vertices
MATH 306 HOMEWORK 5
Let G be a graph and X V (G). We denote by (X ) the set of all edges of G with one
end in X and the other in V (G) X . If D E (G), D is called a cut if D = (X ) for some
X V (G) with = X = V (G).
1. Show that if C is a cycle of G and D
Chapter 5
Graph Embeddings
When considering a graph G, a diagram of G is often drawn (in the plane). Sometimes no edges cross in a drawing, while on other occasions some pairs of edges
may cross. Even if some pairs of edges cross in a diagram of G, there
Subdivisions of Large Complete Bipartite Graphs
and Long Induced Paths in k -Connected Graphs
Thomas Bhme
o
Bojan Mohar
Institut fr Mathematik
u
Technische Universitt Ilmenau
a
Ilmenau, Germany
Department of Mathematics
University of Ljubljana
Ljubljana,
Chapter 6
Introduction to Vertex
Colorings
There is little doubt that the best known and most studied area within graph theory
is coloring.
Graph coloring is arguably the most popular subject in graph theory.
Noga Alon (1993)
The remainder of this book is
Budapest Semesters in Mathematics
ADVANCED COMBINATORICS HANDOUTS
Andrs Gyrfs
a
aa
2011 FALL
INTRODUCTION
This handout is based on the Advanced Combinatorics course I have taught through
many years at Budapest Semesters. It contains a basic material cover
Fishers Theorem
k
Fix a simple digraph D = (V, E ), let v V , and let k Z. If k 0 we let ND (v ) denote
k
the set of vertices at distance k from v , and if k < 0 we let ND (v ) denote the set of vertices
k
with distance k to v . We dene d eg k (v ) = |ND
Pretty Theorems on Vertex Transitive Graphs
Growth
For a graph G a vertex x and a nonnegative integer n we let B (x, n) denote the ball of radius
n around x (i.e. the set cfw_u V (G) : dist (u, v ) n. If G is a vertex transitive graph then
|B (x, n)| = |B
The Gallai-Edmonds Decomposition
Here we present Kotlovs proof of the Gallai-Edmonds decomposition. For every graph G, we
let o dd(G) denote the number of odd components of G (i.e., components H G with |V (H )|
odd) . The following famous theorem of Tutte
Lecture 11
The goal of this lecture is to prove that the size-Ramsey number for the path Pt with t edges is linear
in t. That is, there exists a constant c such that r(Pt ) ct.
We begin with a well-known lemma of Psa. To state it, suppose that, for a give
MATH 306 HOMEWORK 2
1. Let G be a graph with |E (G)| = |V (G)| 1. Show that G is connected if and only if
G has no cycles.
2. Let T be a tree, and let T1 , T2 be connected subgraphs of T with V (T1 T2 ) = .
Show that T1 T2 and T1 T2 are trees.
3. Let T be
MATH 306 FINAL
Turn in by 5pm, Tuesday May 17, 2011, to Paul Seymour in Fine 201 (put it under
the door if necessary) or to one of the TAs. The Princeton honor code applies. You can
use books and lecture notes, and you can use without proof any theorem pr
MATH 306 HOMEWORK 3
1. X E (G) is even-degree if every vertex of G is incident with an even number
of non-loop edges in X . Show that if X and Y are both even-degree then so is
(X Y ) (Y X ). Deduce that if T is a spanning tree of G, there is an even-degr
MATH 306: LECTURE NOTES 7
16. Edge-colorings
A k -edge-coloring of a loopless graph G means a map : E (G) cfw_1, ., k such that
(e) = (f ) for all edges e = f which share an end. The minimum k such that G has a
k -edge-coloring is the edge-chromatic numbe
MATH 306: LECTURE NOTES 6
15. Stable Sets and Cliques
X V (G) is stable if no edge has both ends in X . X is a clique if every two vertices
in X are adjacent.
(15.1) (Ramsey) For any two integers s, t 1 there exists an integer R 0 so that for
every simple
MATH 306: LECTURE NOTES 5
11. Matchings and Tuttes Theorem
A matching M covers a vertex v if v is incident with an edge in M . A matching is
perfect (or complete, or a 1-factor) if it covers every vertex of the graph.
If G is a graph and X V (G), odd (X )
MATH 306: LECTURE NOTES 4
8. Matchings in bipartite graphs
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 the same vertex. If M is a matching in G, a path P of G
is M -alternating if the edges o
MATH 306: LECTURE NOTES 3
5. Shortest Paths
Let s, t be vertices of a connected graph G, and for each edge e let w(e) 0 be its
length. We wish to nd the path P of G from s to t with shortest length, that is, with
eE (P ) w (e) minimum. We write w (P ) for
MATH 306 LECTURE NOTES 2
3. Trees, Forests
Forest: Graph with no cycles.
Tree: Non-null connected forest.
(3.1) If G is a forest then comp(G) = |V (G)| |E (G)|.
Proof: Induction on |E (G)|, using (2.7).
The result of deleting e, when e is an edge, is the
MATH 306: LECTURE NOTES 1
1. Basic Denitions
Graph G: Consists of a set of vertices (denoted by V (G), a set of edges (denoted by E (G),
and an incidence relation; each edge is incident with either one or two vertices (its ends).
Adjacent: Distinct vertic
MATH 306 HOMEWORK 7
1. Show that every 3-edge-connected cubic graph has a perfect matching. (Hint: if
X V (G), | (X )| 3|X |.) What about 2-edge-connected cubic graphs?
2. Let G be a graph and Z V (G). Show that the following are equivalent:
(i) G has a m
MATH 306 HOMEWORK 9
1. Let G be a 2-connected loopless graph drawn in the plane. For each vertex v , dene
1
S (v ) = 1 deg(v ) . Show that for some region r , v r S (v ) < 1, where the sum is
2
over all vertices v incident with r . [Hint: what is the aver
MATH 306 HOMEWORK 10
1. How many cubic graphs G are there (up to isomorphism) such that |V (G)| = 10
and every cycle of G has length 5? Let G1 be the line graph of K5 , let G2 be the
complement of G1 and let G3 be the line graph of G2 . What is the chroma
MATH 306 HOMEWORK 8
1. What is the maximum possible number of edges in a simple planar bipartite graph
with n 3 vertices? Deduce that K3,3 is not planar.
2. Let G be non-null, simple and planar, with no vertex of degree 4. Is it true that G
must have two
MATH 306 HOMEWORK 6
1. Let G be a digraph and for each edge e let (e) 0 be an integer, so that for every
vertex v ,
(e) =
e (v )
(e)
e + (v )
Show there is a list C1 , ., Cn of directed cycles (possibly with repetition) so that for
every edge e of G,
|cfw
Lecture 10
The size-Ramsey number r(H ) is the smallest natural number m such that there exists a graph G
with m edges which is Ramsey with respect to H , that is, such that any two-colouring of the edges of
G produces a monochromatic copy of H . For H =
Lecture 9
A graph H is said to be p-arrangeable if there is an ordering v1 , . . . , vn of the vertices of H such that,
for every vertex vi , the set of left neighbours of the set of right neighbours of vi has size at most p.
The following theorem, genera