Graph theory
Chapter 1
: Introduction
Definitions:
Order
Number of vertices
Size
Number of edges
Induced subgraph
A subgraph F of a graph G whenever u and v are vertices of F and uv
is an
edge in G, then it is an edge in F.
Trail
Walk with no edge traverse
d more than once.
Path
Walk with no vertice traversed more than once.
Circuit
Closed trail with length 3 or more.
Geodesic
A path from u to v with length d(u,v).
Diameter
The greatest distance between any two vertices in a graph.
Complete
A graph is comple
te if every two distinct vertices of G are adjacent.
Complement of graph
The complement has every edge graph G does not have
Bipartite graph
A graph of 2 components, each component is a partite set
Complete bipartite
If every vertice of a component is conn
ected to the vertices of the other
component
k

partite graph
A graph of k components
Join
The union between a graph G en H where all vertices are connected to the
vertices of the other graph
Cartesian product
2 graphs multiplied
Theorems:
Theorem 1.6:
I
f a graph G contains a u

v walk of length l, then
G contains a u

v path of length
at most l
Theorem 1.7:
Let R be the relation defined on the vertex set of a praph G by u R v, where u,v є
V(G) such that u R v and u R w. Hence G contains a u

v path P’ and a
v

w path P’’.
As we have seen earlier, following P’ by P’’ produces a u

w walk W. By Theorem
1.6, G contains a u

w path and so u R w.
Theorem 1.8:
Let G be a graph of order 3 or more. If G contains two distinct vertices u and v
such G

u and G

v are connec
ted, then G itself is connected
Theorem 1.9:
If G is a connected graph of order 3 or more, then G contains two distinct
vertices u and v such that G

u and G

v are connected.
Theorem 1.10:
Let G be a graph of order 3 or more. Then G is connected if and only
if G
contains two distinct vertices u and v such that G

u and G

v are connected
Theorem 1.11:
If G is a disconnected graph, then the complement of G is connected
Theorem 1.12:
A non

trivial graph G is a bipartite graph if and only if G contains no odd cyc
les
Chapter 2: Degrees
Definitions:
Degree of a vertex
Number of edges incident with an edge
Neighbors
Two adjacent vertices
Neighborhood
All neighbors
Even (odd) vertex
Vertex with even(odd) degree
Sharpness
The sharpness of a theory means that some
statement could is not the bound of the statement,
for example bla ≤ n

1 is true, but is bla≤ n

2 also true.
r

regular graph
A graph where every vertex has the degree r
Petersen graph
A 3

regular graph of degree 10(see picture)
Harary graphs
r

regular gr
aphs H
r,n
with order n and the properties of theorem 2.6
Degree sequence
All degrees of the vertices of a graph sorted in a sequence.
Graphical
A degree sequence is graphical if a graph can be constructed from it
Adjacency matrix
Matrix with n columns and
n rows with where
A
ij
=
Incidence matrix
n x m matrix where
B
ij
=
Equal walks
Two walks are equal if they are equal term to term
Theorems:
Theorem 2.1
If G is a graph of size m, then
Corollary 2.3
Every graph has a
n even number of odd vertices
Theorem 2.4
Let G be a graph of order n. If
for every two non

adjacent vertices u and v of G, then G is connected and
diam(G) ≤ 2.
Corollary 2.5
If G is a graph of order n with δ(G) ≥ (n

1)/2, then G is c
onnected
Theorem 2.6
Let r and n be integers with 0 ≤ r ≤ n

1. There exists an r

regular graph of
order n if and only if at least one of r and n are even.
Theorem 2.7
For every graph G and every integer r ≥ Δ(G), there exists an r

regular graph H
containin
g G as an induced subgraph.
Theorem 2.10
A non

increasing sequence s : d
1
, d
2
,…., d
n
(n≥2) of non

negative integers,
where d
1
≥1, is graphical if and only if the sequence
s
1
: d
2
–
1, d
3
–
1, d
d1+1

1, d
d1 + 2
,…d
n
is graphical.
Theorem 2.13
Let G be a
graph with vertex set V(G) = {v
1
, v
2
, … , v
n
} and adjacency matrix A
= [
a
ij
]
. Then the entry a
ij
(k)
in row i and column j of A
k
is the number of distinct
v
i
–
v
j
walks of length k in G.
Chapter 4: Trees
Definitions:
Bridge
An edge uv is a brige when G
–
uv is disconnected, while G is connected
Tree
A tree is an acyclic graph
Caterpillar
A tree of order 3 or more
Spine
A caterpillar without end

vertices
Forest
A cyclic graph where each components is a tree
Isomorphic Graphs
G
raphs t
hat are struct
urally equivalent
Spanning subgraph
Subgraph H is a spanning subgraph of G if it contains every vertex of G.
Spanning tree
If H is a tree it is called a spanning tree
Cost/weight
Edge can have certain costs or weights
Weight of a graph
The weight of a g
raph is all weights summed up
Minimum spanning tree: Spanning tree with minimum weight
Algorithms for mst

problem:

Kruskal’s algorithm:
For a connected weighted graph G, a spanning tree T of G constructed follows: For the first
edge e
1
of T, we select any
edge of G of minimum weight and for the second edge e
2
of T, we
select any remaining edge of minimum weight. For the third edge e
3
of T, we choose any
remaining edge of G of minimum weight that does not produce a cycle with the previously
selected edges. W
e continue in this manner until a spanning tree is produced.

Prim’s algorithm:
For a connected weighted graph G, a spanning tree T of is constructed as follows: For an
arbitrary vertex u for G, an edge minimum weight incident with u is selected as first ed
ge e
1
of T. For subsequent edges e
2
, e
3
,..,e
n

1
, we select an edge of minimum weight among those
edges having exactly one of its vertices incident with an edge already selected.
Theorems:
Theorem 4.1
An edge e of a graph G is a bridge if and only if e l
ies on no cycle of G
Theorem 4.2
A graph G is a tree if and only if every two vertices of G are connected by a
unique path
Theorem 4.3
Every nontrivial tree has at least two end

vertices
Theorem 4.4
Every tree of order n has size n

1
Corollary 4.6
Every fo
rest of order n with k components has size n
–
k
Theorem 4.7
The size of every connected graph of order n is at least n

1
Theorem 4.8
Let G be a graph of order n and size m. If G satisfies any two properties:
1.
G is connected,
2.
G is acyclic,
3.
M = n

1,
then G i
s a tree.
Theorem 4.9
Let T be a tree of order k. If G is a graph with δ(G) ≥ k

1, then T is isomorphic
to some subgraph of G
Theorem 4.10
Every connected graph contains a spanning tree
Theorem 4.11
Kruskal’s algorithm produces a minimum spanning tree in a connecte
d
weighted graph.
Chapter 6 Traversability
Definitions:
Eulerian circuit
A circuit C in a graph G which contains every edge exactly once
Eulerian graph
A graph with an eulerian circuit
Eulerian trail
A trail that contains all edges in a graph
Hamiltonian
cycles
A cycle that contains all vertices in a graph
Hamiltonian graph
A graph that contains a Hamiltonian cycle
Hamiltonian path
A path that contains all vertices in a graph
Closure; C(g)
A graph G with order n, the closure of G is the graph obtained by
recursively
joining pairs of non

adjacent, whose degree sum is at least n, until no such
pair remains.
Cut

vertex
A vertex in a connected graph G if G
–
v is disconnected
Theorems:
Theorem 6.1
A nontrivial connected graph G is Eulerian if and only if ever
y vertex has an
even degree.
Corollary 6.2
A connected graph G contains an Eulerian trail if and only if exactly two
vertices of G have odd degrees. Furthermore, each Eulerian trail of G begins
or ends at one of these odd vertices and ends at the other
The
orem 6.3
Let G and H be nontrivial connected graphs. Then G x H is Eulerian if and only
if both G and H are Eulerian or every vertex of G and H is odd.
Theorem 6.4
The Petersen graph is not Hamiltonian
Theorem 6.5
If G is a Hamiltonian graph, then for ev
ery nonempty proper set S of vertices
of G,
k
(G
–
S
) ≤ S
.
(k(G) is the number of components in G)
Theorem 6.6
Let G be a graph of order n ≥ 3.
If
Deg u + deg v ≥ n
for each pair u, v of nonadjacent vertices of G, then G is Hamiltonian.
Corollary 6.7
Let G be a graph of order n ≥ 3. If deg v ≥ n/2 f
or each vertex v of G, the G is
Hamiltonian.
Theorem 6.8
Let u and v be nonadjacent vertices in a graph G of order n such that deg u +
deg v ≥ n. Then G + uv is Hamiltonian if and only if G is Hamiltonian.
Theorem 6.9
A graph is Hamiltonian if and only if
its closure is Hamiltonian
Corollary 6.10
If G is a graph of order at least 3 such that C(G) is complete, then G is
Hamiltonian
Theorem 6.11
Let G be a graph of order n ≥ 3. If for every integer j with 1 ≤ j ≤
, the
number of vertices of G w
ith degree at most j is less than j, then G is
Hamiltonian.
Chapter 8
Matchings and factorization
Definitions:
Independent
A set of edges
(vertices)
is independent if no two edges
(vertices)
are adjacent
Matching
Let G be a bipartite graph with partite set
s U and W, where r = U ≤ W. A
matching in G is therefore a set M = {e
1
,e
2
,…,e
k
} of edges, where e
i
= u
i
w
i
for 1
≤ i ≤ k such that u
1
, u
2
, … u
k
are k distinct vertices of U and w
1
, w
2
, … , w
k
are k
distinct vertices of W.
Neighborly
Given neighborhood
N(X) a set U is neighborly if N(X)≥ X where for every
nonempty subset X of U
and N(X) the union of all neighbors x є X
.
If the
neighborhood of a subset of X of U is bigger or equal to the subset X for
every
subset of X of U, then U is neighborly.
Cove
r
A vertex and an incident edge cover each other
Edge cover
A graph without isolated vertices is a set of edges of G that covers all vertices
of G
α(G)
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M楮業um mb敲eo映v敲瑩e敳e瑨慴
捯v敲e慬a 敤e敳eo映f
β
(G)
Vertex independence
number
Maximum number of vertices, no
two which are adjacent
α
1
(G)
Edge covering number
Minimum number of edges that
cover all vertices of G
β
1
(G)
Edge independence number
Maximum number of edges, no
two which are adjacent
Theorems:
Theorem 8.3
Let G be a bipartite graph with partite sets U and W such that r = U ≤ W.
Then G contains a matching of cardinality r if and only if U is neighborly.
Theorem 8.4
A collection {S
1
, S
2
, … , S
n
} of a nonempty finite sets has a system of distin
ct
representatives if and only if for each integer k with 1 ≤ k ≤ n, the union of
any k of these sets contains at least k elements.
Theorem 8.5
In a collection of r women and s men, where 1 ≤ r ≤ s, a total of r marriages
(Marriage Theorem)
between acquai
nted couples is possible if and only if for each integer k with
1 ≤ k ≤ r, every subset of k women is collectively acquainted with at least k
men.
Theorem 8.6
Every r

regular bipartite graph (r ≥ 1) has a perfect matching
Theorem 8.7
For every graph G of
order n containing no isolated vertices,
α
1
(G) +
β
1
(G) = n
.
Theorem 8.8
For every graph G of order n containing no isolated vertices,
α
(G) +
β
(G) = n.
8.14
A graph G without isolated vertices contains a has a perfect a perfect
matching if and o
nly if
α
1
(G) =
β
1
(G)
.
For 8.2 see handout Hungarian method
Chapter 9 Planarity
Definitions:
Planar graph
A graph that can we drown without two edges crossing each other
Plane graph
A graph is drawn in a plane without no two edges of G cross
Regions
The
connected pieces of a plane graph
Exterior region
The unbounded part of every plane graph
Boundary of a region
The subgraph of all vertices and edges that are
incident with the region
Maximal planar
G is planar, but the addition of an edge between
two non
adjacent vertices of G makes it nonplanar
Subdivision
One or more vertices of degree 2 are inserted
into one or more edges of G(see picture)
Theorems:
Theorem 9.1
If G is a connected plane graph of order n, size m, and having r regions, then
n
–
m + r = 2
Theorem 9.2
If G is a planar graph of order n ≥ 3 and size m, then
m ≤ 3n
–
6.
Corollary 9.3
Every planar graph contains a vertex of
degree 5 or less
Corollary 9.4
The complete graph K
5
is nonplanar
(see
picture)
Theorem 9.5
The graph K
3,3
is nonpla
nar
(see picture)
Theorem 9.7
A graph G is planar if and only if G does not contain K
5
, K
3,3
or a subdivision of
K
5
or K
3,3
as a subgraph
.
Chapter 10 Coloring
Definitions:
Dual
If
you represent a map by a graph it is called a dual
Coloring
Assigning col
ors to regions, such that every adjacent vertex has a separate
color
Chromatic number
Smallest number of colors needed for coloring denoted as χ(G)
k

coloring
If it is possible to color G from a set of k colors, then G is said to be k

colorable
k

coloring
A coloring that uses k colors
Color classes
if G is k

chromatic then it is possible to divide G in k independent sets, these
are called color classes
Clique
A complete subgraph of graph G.
Clique number
The order of the largest clique, denoted as ω(G)
Sha
dow graph
Obtained from graph G by adding, for each vertex v of G, a new vertex v’,
called the
shadow vertex
of v, and joining v’ to the neighbors of v in G
.
Theorems:
Theorem 10.1
The chromatic number of every planar graph is at most 4(The Four Color
Th
eorem)
Theorem 10.2
A graph G has chromatic number 2 if and only if G is a nonempty bipartite
graph
Theorem 10.5
For every graph G of order n:
χ(G) ≥ ω(G) and χ(G) ≥
Theorem 10.7
For every graph:
χ(G) ≤ 1 + Δ(G)
Theorem 10.8
For every co
nnected graph G that is not an odd cycle or a complete graph
χ(G) ≤ Δ(G)
Theorem 10.9
For every graph G,
χ(G) ≤ 1 + max{δ(H)},
where the maximum is taken over all induced subgraphs H of G.
Theorem 10.10
For every integer k ≥ 3, there exists a tr
iangle

free graph with chromatic
number k.
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