graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs...
Transcript of graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs...
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GRAPH THEORY
Yijia ChenShanghai Jiaotong University
2008/2009
Shanghai
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GRAPH THEORY (I) Page 2
Textbook
Shanghai
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GRAPH THEORY (I) Page 2
Textbook
Reinhard Diestel. Graph Theory, 3rd Edition, Spinger, 2005.
Shanghai
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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
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GRAPH THEORY (I) Page 3
Some Requirements
Shanghai
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GRAPH THEORY (I) Page 3
Some Requirements
- QUIET!
Shanghai
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GRAPH THEORY (I) Page 3
Some Requirements
- QUIET!
- MATHEMATICAL RIGOR.
Shanghai
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GRAPH THEORY (I) Page 4
Chapter 1. The Basics
Shanghai
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GRAPH THEORY (I) Page 5
1.1 Graphs
Shanghai
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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
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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.
Shanghai
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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.
Shanghai
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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.
Shanghai
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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).
Shanghai
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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
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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.
G can be finite, infinite, or countable according to its order |G|.
Shanghai
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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.
G can be finite, infinite, or countable according to its order |G|. Unless otherwise stated, ourgraphs will be finite.
Shanghai
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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.
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.
Shanghai
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GRAPH THEORY (I) Page 7
1.2 The degree of a vertex
Shanghai
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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.
Shanghai
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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
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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).
A vertex v ∈ V (G) is incident with an edge e ∈ E(G), if v ∈ e; then e is an edge at v.
Shanghai
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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).
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
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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).
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.
Shanghai
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GRAPH THEORY (I) Page 8
A vertex if degree 0 is isolated.
Shanghai
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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}
.
Shanghai
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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
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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}
.
If δ(G) = Δ(G) = k, then G is k-regular, or simply regular.
Shanghai
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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}
.
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).
Shanghai
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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 |.
Shanghai
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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.
Shanghai
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GRAPH THEORY (I) Page 10
Let G = (V, E) be a graph. Then ε(G) := |E|/|V |.
Shanghai
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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).
Shanghai
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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
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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.
If E′ ={xy ∈ E
∣∣ x, y ∈ V ′}, then G′ is an induced subgraph of G, written as
G′ = G[V ′].
Shanghai
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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.
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).
Shanghai
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GRAPH THEORY (I) Page 12
1.3 Paths and cycles
Shanghai
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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
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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.
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
![Page 40: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/40.jpg)
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
![Page 41: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/41.jpg)
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Shanghai
![Page 42: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/42.jpg)
GRAPH THEORY (I) Page 13
Why “the” path of length k?
Isomorphic graphs
Shanghai
![Page 43: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/43.jpg)
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
![Page 44: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/44.jpg)
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
![Page 45: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/45.jpg)
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
![Page 46: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/46.jpg)
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′.
Shanghai
![Page 47: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/47.jpg)
GRAPH THEORY (I) Page 14
We often refer to a path by the natural sequence of its vertices, writing, say P = x0x1 . . . xk.
Shanghai
![Page 48: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/48.jpg)
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
![Page 49: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/49.jpg)
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
![Page 50: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/50.jpg)
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.
Shanghai
![Page 51: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/51.jpg)
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).
Shanghai
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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.
Shanghai
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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
![Page 54: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/54.jpg)
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
![Page 55: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/55.jpg)
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.
Shanghai
![Page 56: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/56.jpg)
GRAPH THEORY (I) Page 17
Proposition. Every graph G containing a cycle satisfies g(G) ≤ 2diam G + 1.
Shanghai
![Page 57: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/57.jpg)
GRAPH THEORY (I) Page 18
A vertex is central in G if its greatest distance from any other vertex is as small as possible.
Shanghai
![Page 58: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/58.jpg)
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
![Page 59: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/59.jpg)
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.
Shanghai
![Page 60: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/60.jpg)
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.
Shanghai
![Page 61: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/61.jpg)
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.
�
Shanghai
![Page 62: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/62.jpg)
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
![Page 63: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/63.jpg)
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.
Shanghai
![Page 64: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/64.jpg)
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).
Shanghai
![Page 65: graph1 - SJTUbasics.sjtu.edu.cn/~chen/teaching/GR08/graph1.pdf · GRAPH THEORY(I) Page 5 1.1 Graphs A graphis a pair G =(V,E)of sets such that E ⊆ [V]2. note: For any set A, we](https://reader033.fdocuments.in/reader033/viewer/2022050114/5f4b561db917676e6335b58c/html5/thumbnails/65.jpg)
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