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GRAPH THEORY
Yijia ChenShanghai Jiaotong University
2008/2009
Shanghai
GRAPH THEORY (I) Page 2
Textbook
Shanghai
GRAPH THEORY (I) Page 2
Textbook
Reinhard Diestel. Graph Theory, 3rd Edition, Spinger, 2005.
Shanghai
GRAPH THEORY (I) Page 2
Textbook
Reinhard Diestel. Graph Theory, 3rd Edition, Spinger, 2005.
Available at:
http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
Shanghai
GRAPH THEORY (I) Page 3
Some Requirements
Shanghai
GRAPH THEORY (I) Page 3
Some Requirements
- QUIET!
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GRAPH THEORY (I) Page 3
Some Requirements
- QUIET!
- MATHEMATICAL RIGOR.
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GRAPH THEORY (I) Page 4
Chapter 1. The Basics
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GRAPH THEORY (I) Page 5
1.1 Graphs
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GRAPH THEORY (I) Page 5
1.1 Graphs
A graph is a pair G = (V, E) of sets such that E ⊆ [V ]2.
Shanghai
GRAPH THEORY (I) Page 5
1.1 Graphs
A graph is a pair G = (V, E) of sets such that E ⊆ [V ]2.
note: For any set A, we use [A]k to denote the set of all k-element subsets of A.
Shanghai
GRAPH THEORY (I) Page 5
1.1 Graphs
A graph is a pair G = (V, E) of sets such that E ⊆ [V ]2.
note: For any set A, we use [A]k to denote the set of all k-element subsets of A.
It implies that our graphs are simple, i.e., without self-loops and multiple edges.
Shanghai
GRAPH THEORY (I) Page 5
1.1 Graphs
A graph is a pair G = (V, E) of sets such that E ⊆ [V ]2.
note: For any set A, we use [A]k to denote the set of all k-element subsets of A.
It implies that our graphs are simple, i.e., without self-loops and multiple edges.
- We shall always assume that V ∩ E = ∅.
- The elements of V are the vertices of the graph G, and the elements of E are its edges.
Shanghai
GRAPH THEORY (I) Page 5
1.1 Graphs
A graph is a pair G = (V, E) of sets such that E ⊆ [V ]2.
note: For any set A, we use [A]k to denote the set of all k-element subsets of A.
It implies that our graphs are simple, i.e., without self-loops and multiple edges.
- We shall always assume that V ∩ E = ∅.
- The elements of V are the vertices of the graph G, and the elements of E are its edges.
Let G be a graph. The vertex set is also referred to as V (G) and edge set as E(G).
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GRAPH THEORY (I) Page 6
Let G be a graph. Its order |G| := |V (G)|, i.e., the number of its vertices.
Shanghai
GRAPH THEORY (I) Page 6
Let G be a graph. Its order |G| := |V (G)|, i.e., the number of its vertices.
G can be finite, infinite, or countable according to its order |G|.
Shanghai
GRAPH THEORY (I) Page 6
Let G be a graph. Its order |G| := |V (G)|, i.e., the number of its vertices.
G can be finite, infinite, or countable according to its order |G|. Unless otherwise stated, ourgraphs will be finite.
Shanghai
GRAPH THEORY (I) Page 6
Let G be a graph. Its order |G| := |V (G)|, i.e., the number of its vertices.
G can be finite, infinite, or countable according to its order |G|. Unless otherwise stated, ourgraphs will be finite.
G is the empty graph if |G| = 0, hence V (G) = E(G) = ∅.
G is a trivial graph if |G| ≤ 1.
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GRAPH THEORY (I) Page 7
1.2 The degree of a vertex
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GRAPH THEORY (I) Page 7
1.2 The degree of a vertex
Let G be a graph. Two vertices x, y ∈ V (G) are adjacent, or neighbours, if {x, y} ∈ E(G)or xy is an edge in G.
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GRAPH THEORY (I) Page 7
1.2 The degree of a vertex
Let G be a graph. Two vertices x, y ∈ V (G) are adjacent, or neighbours, if {x, y} ∈ E(G)or xy is an edge in G.
The set of neighbours of a vertex v in G is denoted by NG(v), or N(v).
Shanghai
GRAPH THEORY (I) Page 7
1.2 The degree of a vertex
Let G be a graph. Two vertices x, y ∈ V (G) are adjacent, or neighbours, if {x, y} ∈ E(G)or xy is an edge in G.
The set of neighbours of a vertex v in G is denoted by NG(v), or N(v).
A vertex v ∈ V (G) is incident with an edge e ∈ E(G), if v ∈ e; then e is an edge at v.
Shanghai
GRAPH THEORY (I) Page 7
1.2 The degree of a vertex
Let G be a graph. Two vertices x, y ∈ V (G) are adjacent, or neighbours, if {x, y} ∈ E(G)or xy is an edge in G.
The set of neighbours of a vertex v in G is denoted by NG(v), or N(v).
A vertex v ∈ V (G) is incident with an edge e ∈ E(G), if v ∈ e; then e is an edge at v. The
set of all the edges at v is denoted by E(v).
Shanghai
GRAPH THEORY (I) Page 7
1.2 The degree of a vertex
Let G be a graph. Two vertices x, y ∈ V (G) are adjacent, or neighbours, if {x, y} ∈ E(G)or xy is an edge in G.
The set of neighbours of a vertex v in G is denoted by NG(v), or N(v).
A vertex v ∈ V (G) is incident with an edge e ∈ E(G), if v ∈ e; then e is an edge at v. The
set of all the edges at v is denoted by E(v).
The degree (or valence) dG(v) = d(v) of a vertex v us the number |E(v)| of edges at v,
which is equal to the number of neighbours of v.
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GRAPH THEORY (I) Page 8
A vertex if degree 0 is isolated.
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GRAPH THEORY (I) Page 8
A vertex if degree 0 is isolated.
minimum degree of G is δ(G) := min{d(v)
∣∣ v ∈ V}
.
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GRAPH THEORY (I) Page 8
A vertex if degree 0 is isolated.
minimum degree of G is δ(G) := min{d(v)
∣∣ v ∈ V}
.
maximum degree of G is Δ(G) := max{d(v)
∣∣ v ∈ V}
.
Shanghai
GRAPH THEORY (I) Page 8
A vertex if degree 0 is isolated.
minimum degree of G is δ(G) := min{d(v)
∣∣ v ∈ V}
.
maximum degree of G is Δ(G) := max{d(v)
∣∣ v ∈ V}
.
If δ(G) = Δ(G) = k, then G is k-regular, or simply regular.
Shanghai
GRAPH THEORY (I) Page 8
A vertex if degree 0 is isolated.
minimum degree of G is δ(G) := min{d(v)
∣∣ v ∈ V}
.
maximum degree of G is Δ(G) := max{d(v)
∣∣ v ∈ V}
.
If δ(G) = Δ(G) = k, then G is k-regular, or simply regular.
average degree of G is
d(G) :=1|V |
∑v∈V
d(v).
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GRAPH THEORY (I) Page 9
Proposition. Let G = (V, E) be a graph. Then
|E| =12
∑v∈V
d(v) =12d(G) · |V |.
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GRAPH THEORY (I) Page 9
Proposition. Let G = (V, E) be a graph. Then
|E| =12
∑v∈V
d(v) =12d(G) · |V |.
Proposition. The number of vertices of odd degree in a graph is always even.
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GRAPH THEORY (I) Page 10
Let G = (V, E) be a graph. Then ε(G) := |E|/|V |.
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GRAPH THEORY (I) Page 10
Let G = (V, E) be a graph. Then ε(G) := |E|/|V |.
Proposition. ε(G) = 12d(G).
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GRAPH THEORY (I) Page 11
Subgraphs
Let G = (V, E) and G′ = (V ′, E′) be two graphs. If V ′ ⊆ V and E′ ⊆ E, then G′ is a
subgraph of G (and G a supergraph of G′), written as G′ ⊆ G.
Shanghai
GRAPH THEORY (I) Page 11
Subgraphs
Let G = (V, E) and G′ = (V ′, E′) be two graphs. If V ′ ⊆ V and E′ ⊆ E, then G′ is a
subgraph of G (and G a supergraph of G′), written as G′ ⊆ G.
If E′ ={xy ∈ E
∣∣ x, y ∈ V ′}, then G′ is an induced subgraph of G, written as
G′ = G[V ′].
Shanghai
GRAPH THEORY (I) Page 11
Subgraphs
Let G = (V, E) and G′ = (V ′, E′) be two graphs. If V ′ ⊆ V and E′ ⊆ E, then G′ is a
subgraph of G (and G a supergraph of G′), written as G′ ⊆ G.
If E′ ={xy ∈ E
∣∣ x, y ∈ V ′}, then G′ is an induced subgraph of G, written as
G′ = G[V ′].
Proposition. Every graphs G with at least one edge has a subgraph with
δ(H) > ε(H) ≥ ε(G).
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GRAPH THEORY (I) Page 12
1.3 Paths and cycles
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GRAPH THEORY (I) Page 12
1.3 Paths and cycles
A path is a non-empty graph P = (V, E) of the form
V = {x0, x1, . . . , xk} and E = {x0x1, x1x2, . . . , xk−1xk},
where the xi are all pairwise distinct.
Shanghai
GRAPH THEORY (I) Page 12
1.3 Paths and cycles
A path is a non-empty graph P = (V, E) of the form
V = {x0, x1, . . . , xk} and E = {x0x1, x1x2, . . . , xk−1xk},
where the xi are all pairwise distinct.
The vertices x0 and xk are linked by P and are called its ends; the vertices x1, . . . , xk−1 are
the inner vertices of P .
The number of edges of a path is its length, and the path of length k is denoted by Pk.
Shanghai
GRAPH THEORY (I) Page 12
1.3 Paths and cycles
A path is a non-empty graph P = (V, E) of the form
V = {x0, x1, . . . , xk} and E = {x0x1, x1x2, . . . , xk−1xk},
where the xi are all pairwise distinct.
The vertices x0 and xk are linked by P and are called its ends; the vertices x1, . . . , xk−1 are
the inner vertices of P .
The number of edges of a path is its length, and the path of length k is denoted by Pk. note:
k is allowed to be zero.
Shanghai
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Shanghai
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Isomorphic graphs
Shanghai
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Isomorphic graphs
Let G = (V, E) and G′ = (V ′, E′) be two graphs. We call G and G′ isomorphic, and write
G � G′, if there exists a bijection ϕ : V → V ′ such that
xy ∈ E ⇐⇒ ϕ(x)ϕ(y) ∈ E′
for all x, y ∈ V .
Shanghai
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Isomorphic graphs
Let G = (V, E) and G′ = (V ′, E′) be two graphs. We call G and G′ isomorphic, and write
G � G′, if there exists a bijection ϕ : V → V ′ such that
xy ∈ E ⇐⇒ ϕ(x)ϕ(y) ∈ E′
for all x, y ∈ V .
- ϕ is an isomorphism.
Shanghai
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Isomorphic graphs
Let G = (V, E) and G′ = (V ′, E′) be two graphs. We call G and G′ isomorphic, and write
G � G′, if there exists a bijection ϕ : V → V ′ such that
xy ∈ E ⇐⇒ ϕ(x)ϕ(y) ∈ E′
for all x, y ∈ V .
- ϕ is an isomorphism.
- If G = G′, then ϕ is an automorphism.
Shanghai
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Isomorphic graphs
Let G = (V, E) and G′ = (V ′, E′) be two graphs. We call G and G′ isomorphic, and write
G � G′, if there exists a bijection ϕ : V → V ′ such that
xy ∈ E ⇐⇒ ϕ(x)ϕ(y) ∈ E′
for all x, y ∈ V .
- ϕ is an isomorphism.
- If G = G′, then ϕ is an automorphism.
- We do not normally distinguish between isomorphic graphs, and write G = G′ instead of
G � G′.
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GRAPH THEORY (I) Page 14
We often refer to a path by the natural sequence of its vertices, writing, say P = x0x1 . . . xk.
Shanghai
GRAPH THEORY (I) Page 14
We often refer to a path by the natural sequence of its vertices, writing, say P = x0x1 . . . xk.
If P = x0 . . . xk−1 is a path and k ≥ 3, then the graph
C := P + xk−1x0
is called a cycle.
Shanghai
GRAPH THEORY (I) Page 14
We often refer to a path by the natural sequence of its vertices, writing, say P = x0x1 . . . xk.
If P = x0 . . . xk−1 is a path and k ≥ 3, then the graph
C := P + xk−1x0
is called a cycle.
The cycle C might be written as x0 . . . xk−1x0.
Shanghai
GRAPH THEORY (I) Page 14
We often refer to a path by the natural sequence of its vertices, writing, say P = x0x1 . . . xk.
If P = x0 . . . xk−1 is a path and k ≥ 3, then the graph
C := P + xk−1x0
is called a cycle.
The cycle C might be written as x0 . . . xk−1x0.
The length of a cycle is its number of edges (or vertices); the cycle of length k is called a
k-cycle and denoted by Ck.
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GRAPH THEORY (I) Page 15
Proposition. Every graph G contains a path of length δ(G) and a cycle of length at least
δ(G) + 1 (provided that δ(G) ≥ 2).
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GRAPH THEORY (I) Page 16
The minimum length of a cycle in a graph G is the girth g(G) of G; the maximum length of
a cyle in G is its circumference.
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GRAPH THEORY (I) Page 16
The minimum length of a cycle in a graph G is the girth g(G) of G; the maximum length of
a cyle in G is its circumference.
If G does not contain a cycle, then its girth is ∞, and its circumference is 0.
Shanghai
GRAPH THEORY (I) Page 16
The minimum length of a cycle in a graph G is the girth g(G) of G; the maximum length of
a cyle in G is its circumference.
If G does not contain a cycle, then its girth is ∞, and its circumference is 0.
The distance dG(x, y) in G of two vertices x, y is the length of a shortest path between x and
y in G; if no such path exists, then we set dG(x, y) := ∞.
Shanghai
GRAPH THEORY (I) Page 16
The minimum length of a cycle in a graph G is the girth g(G) of G; the maximum length of
a cyle in G is its circumference.
If G does not contain a cycle, then its girth is ∞, and its circumference is 0.
The distance dG(x, y) in G of two vertices x, y is the length of a shortest path between x and
y in G; if no such path exists, then we set dG(x, y) := ∞.
The greatest distance between any two vertices in G is the diameter of G, denoted by
diam G.
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GRAPH THEORY (I) Page 17
Proposition. Every graph G containing a cycle satisfies g(G) ≤ 2diam G + 1.
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GRAPH THEORY (I) Page 18
A vertex is central in G if its greatest distance from any other vertex is as small as possible.
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GRAPH THEORY (I) Page 18
A vertex is central in G if its greatest distance from any other vertex is as small as possible.
The distance is the radius of G, denoted by rad G. Formally
rad G := minx∈V (G)
maxy∈V (G)
dG(x, y).
Shanghai
GRAPH THEORY (I) Page 18
A vertex is central in G if its greatest distance from any other vertex is as small as possible.
The distance is the radius of G, denoted by rad G. Formally
rad G := minx∈V (G)
maxy∈V (G)
dG(x, y).
Proposition. rad G ≤ diam G ≤ 2rad G.
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GRAPH THEORY (I) Page 19
Proposition. A graph G of radius at most k and maximum degree at most d ≥ 3 has fewerthan d
d−2 (d − 1)k vertices.
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GRAPH THEORY (I) Page 19
Proposition. A graph G of radius at most k and maximum degree at most d ≥ 3 has fewerthan d
d−2 (d − 1)k vertices.
Proof.
|G| ≤ 1 + dk−1∑i=0
(d − 1)i = 1 +d
d − 2((d − 1)k − 1
)<
d
d − 2(d − 1)k.
�
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GRAPH THEORY (I) Page 20
For d, g ∈ N let
n0(d, g) :=
⎧⎪⎪⎪⎨⎪⎪⎪⎩
1 + dr−1∑i=0
(d − 1)i if g = 2r + 1 is odd;
2r−1∑i=0
(d − 1)i if g = 2r is even.
Shanghai
GRAPH THEORY (I) Page 20
For d, g ∈ N let
n0(d, g) :=
⎧⎪⎪⎪⎨⎪⎪⎪⎩
1 + dr−1∑i=0
(d − 1)i if g = 2r + 1 is odd;
2r−1∑i=0
(d − 1)i if g = 2r is even.
Proposition. A graph of minimum degree δ and girth g has at least n0(δ, g) vertices.
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GRAPH THEORY (I) Page 21
Theorem.[Alon, Hoory and Linial 2002] Let G be a graph. If d(G) ≥ d ≥ 2 and
g(G) ≥ g ∈ N then |G| ≥ n0(d, g).
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GRAPH THEORY (I) Page 21
Theorem.[Alon, Hoory and Linial 2002] Let G be a graph. If d(G) ≥ d ≥ 2 and
g(G) ≥ g ∈ N then |G| ≥ n0(d, g).
Corollary. If δ(G) ≥ 3 then g(G) < 2 log |G|.
Shanghai
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