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GRAPH THEORY

Yijia ChenShanghai Jiaotong University

2008/2009

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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/

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Some Requirements

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Some Requirements

- QUIET!

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Some Requirements

- QUIET!

- MATHEMATICAL RIGOR.

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Chapter 1. The Basics

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1.1 Graphs

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1.1 Graphs

A graph is a pair G = (V, E) of sets such that E ⊆ [V ]2.

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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.

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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.

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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.

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.

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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.

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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Let G be a graph. Its order |G| := |V (G)|, i.e., the number of its vertices.

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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|.

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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.

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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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1.2 The degree of a vertex

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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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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).

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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.

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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).

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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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A vertex if degree 0 is isolated.

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A vertex if degree 0 is isolated.

minimum degree of G is δ(G) := min{d(v)

∣∣ v ∈ V}

.

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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}

.

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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.

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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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Proposition. Let G = (V, E) be a graph. Then

|E| =12

∑v∈V

d(v) =12d(G) · |V |.

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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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Let G = (V, E) be a graph. Then ε(G) := |E|/|V |.

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Let G = (V, E) be a graph. Then ε(G) := |E|/|V |.

Proposition. ε(G) = 12d(G).

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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.

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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 ′].

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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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1.3 Paths and cycles

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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.

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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.

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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.

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Why “the” path of length k?

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Why “the” path of length k?

Isomorphic graphs

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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 .

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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.

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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.

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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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We often refer to a path by the natural sequence of its vertices, writing, say P = x0x1 . . . xk.

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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.

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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.

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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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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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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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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.

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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) := ∞.

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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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Proposition. Every graph G containing a cycle satisfies g(G) ≤ 2diam G + 1.

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A vertex is central in G if its greatest distance from any other vertex is as small as possible.

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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).

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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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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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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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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.

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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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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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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|.

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