Graph theory chapter 2

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1.3. VERTEX DEGREES 11

1.3 Vertex Degrees

Vertex Degree for Undirected Graphs: Let G be an undirected

graph and x ∈ V (G).

The degree dG(x) of x in G: the number of edges incident with x, each loop

counting as two edges.

For the graph G shown in Figure 1.9 (a), for instance,

dG(x1) = dG(x3) = 4, dG(x2) = dG(x4) = 3.

x1

x2

x3x4

e1e2

e3e4

e5

e6

e7

(a)

x1

x2

x3x4

a2

a7 a3a4

a5

a6

a1

(b)

Figure 1.9: (a) an undirected graph G (b) a digraph D

A vertex of degree d is called a d-degree vertex. A 0-degree vertex is called an

isolated vertex. A vertex is called to be odd or even if its degree is odd or even.

A graph G is k-regular if dG(x) = k for each x ∈ V (G), and G is regular if it is

k-regular for some k , and k is called the regularity

of G.For instance, K n is (n−1)-regular; K n,n is n-regular; Petersen graph is 3-regular;

the n-cube is n-regular.

The maximum degree of G: ∆(G) = max{dG(x) : x ∈ V (G)}.

The minimum degree of G: δ (G) = min{dG(x) : x ∈ V (G)}.

Clearly, δ (G) = k = ∆(G) if G is k -regular.

Vertex Degree for Digraphs: Let D be a digraph and y ∈ V (D).

E +

D

(y): a set of out-going edges of y in D .

E −D(y): a set of in-coming edges of y in D .

The out-degree of y : d+D(y) = |E +D(y)|.

The in-degree of y : d−D(y) = |E −D(y)|.

For the digraph D shown in Figure 1.9 (b), for instance,

d+D(y1) = 2, d+

D(y2) = 1, d+D(y3) = 1, d+

D(y4) = 3;d−D(y1) = 2, d−

D(y2) = 2, d−D(y3) = 3, d−

D(y4) = 0,

A vertex y is called to be balanced if d+D(y) = d−

D(y), and D is called to be



12 Basic Concepts of Graphs

balanced if each of its vertices is balanced. The parameters

∆+(D) = max{d+D(y) : y ∈ V (D)}, and

∆−(D) = max{d−D(y) : y ∈ V (D)}

are the maximum out-degree and maximum in-degree of D, respectively. The

parameters

δ +(D) = min{d+D(y) : y ∈ V (D)}, and

δ −(D) = min{d−D(y) : y ∈ V (D)}

are the minimum out-degree and minimum in-degree of D , respectively. The

parameters

∆(D) = max{∆+(D), ∆−(D)}, andδ (D) = min {δ +(D), δ −(D)}

are the maximum and the minimum degree of a digraph D, respectively. A

digraph D is k -regular if ∆(D) = δ (D) = k.

The First Theorem: Let G be a bipartite undirected graph with a bipar-

tite {X, Y }. It is easy to see that the relationship between degree of vertices and

the number of edges of G is as follows.

x∈X

dG(x) = ε(G) =y∈Y

dG(y). (1.3)

As a result, we have

2 ε(G) =

x∈V (G)

dG(x). (1.4)

Generally, for any a digraph D we have the following relationship between degree

of vertices and the number of edges of G.

Theorem 1.1 For any digraph D,

ε(D) =x∈V

d+D(x) =

x∈V

d−D(x).

Proof: Let G be the associated bipartite graph with D of bipartition {X, Y }.

Note that dG(x′

) = d

+

D(x), dG(x′′

) = d−

D(x), ∀ x ∈ V (D). By the equality (1.3),we have that

x∈V

d+D(x) =

x′∈X

dG(x′) = ε(G) =x′′∈Y

dG(x′′) =x∈V

d−D(x).

Since ε(D) = ε(G) by (1.2), the theorem follows.

Corollary 1.1 For any undirected graph G,

2ε(G) =x∈V

dG(x)



1.3. VERTEX DEGREES 13

and the number of vertices of odd degree is even.

Proof: Let D be the symmetric digraph of G. Then ε(D) = 2ε(G). Note that

dG(x) = d+D(x) = d−

D(x), ∀ x ∈ V .

By Theorem 1.1, we have that

x∈V

dG(x) =x∈V

d+D(x) =

x∈V

d−D(x) = ε(D) = 2ε(G).

Let V o and V e be the sets of vertices of odd and even degree in G, respectively.

Thenx∈V o

dG(x) +x∈V e

dG(x) =x∈V

dG(x) = 2ε(G).

Since x∈V e

dG(x) is even, it follows that x∈V o

dG(x) is also even. Since dG(x) is odd

for any x ∈ V o, thus, |V o| is even.

Others Notations: The following notation and terminology are useful and

convenient to our discussions later on.

Let D be a digraph, S and T are disjoint nonempty subset of V (D). The symbol

E D(S, T ) denotes the set of edges of D whose tails are in S and heads are in T , and

µD(S, T ) = |E D(S, T )|. When just one graph is under discussion, we usually omit

the letter D from these symbols and write (S, T ) and µ(S, T ) instead of E D(S, T )

and µD(S, T ) for short. [S, T ] = (S, T )∪ (T, S ). If T = S = V (D)\S , then we write

E +D(S ) (resp. E −D(S )) instead of (S, S ) (resp. (S, S )), and d+D(S ) = |E +D(S )| (resp.

d−D(S ) = |E −D(S )|).

The symbol N +D (S ) (resp. N −D (S )) denotes the set of heads (resp. tails) of edges

in E D[S ], which is called a set of out-neighbors (resp. in-neighbors) S in D .

For instance, consider the digraph D shown in Figure 1.9. Let S = {y1, y2}, then

E +D(S ) = {a3}, d+D(S ) = 1, N +D(S ) = {y3},

E −D(S ) = {a4, a7}, d−D(S ) = 2, N −D (S ) = {y3, y4}.

Similarly, for an undirected graph G and S ⊂ V (G), the symbols E G(S ) and

N G(S ) denote the set of edges incident with vertices in S in G and the set of

neighbors of S in G, dG(S ) = |E G(S )|.

Example 1.3.1 Prove that ε(G) ≤ 14 v2 for any simple undirected graph G

without triangles.

Proof: Arbitrarily choose xy ∈ E (G). Since G is simple and contains no

triangle, it follows that

[dG(x) − 1] + [dG(y) − 1] ≤ v − 2,




that is,

dG(x) + dG(y) ≤ v.

Then summing over all edges in G yields

x∈V

d2G(x) ≤ v ε.

By Cauchy’s inequality and Corollary 1.1, we have that

v ε ≥x∈V

d2

G(x) ≥ 1

v

x∈V

dG(x)

2

= 4

v ε2,

that is, ε(G) ≤ 1

4 v2.

Example 1.3.2 Let G is a self-complementary simple undirected graph with

v ≡ 1 (mod 4). Prove that the number of vertices of degree 1

2 (v − 1) in G is odd

(the self-complementary graph is defined in the exercise 1.2.6).

Proof: Let V o and V e be the sets of vertices of odd and even degree in G,

respectively. Then |V o| is even by Corollary 1.1. Since v ≡ 1 (mod 4), v must be

odd and, thus, |V e| is odd and 1

2 (v − 1) is even. Let

V ′ = {x ∈ V (G) : dG(x) = 1

2 (v − 1)}.

To prove the conclusion, we only need to show that |V ′ | is even. To the end, let

x ∈ V ′ with dGc(x) = 12

(v − 1) since G ∼= Gc. Then there must exist yx ∈ V (G)

with dG(yx) = dGc(x). Note that

dG(yx) = dGc(x) = (v − 1) − dGc(x) = 1

2 (v − 1). (1.5)

Thus, yx = x from (1.5) and yx ∈ V ′ . Furthermore, yx = yz if x, z ∈ V ′ and x = z .

This fact implies that the vertices in V ′ occur in pairs, which shows that |V ′ | is even.



1.4. SUBGRAPHS AND OPERATIONS 15

1.4 Subgraphs and Operations

A subgraph is one of the most basic concepts in graph theory. In this section,

we first introduce various subgraphs induced by operations of graphs.

Subgraphs: Suppose that G = (V (G), E (G), ψG) is a graph. A graph

H = (V (H ), E (H ), ψH ) is called a subgraph of G, denoted by H ⊆ G, or G

is a supergraph of H if V (H ) ⊆ V (G), E (H ) ⊆ E (G) and ψH is the restriction of

ψG to E (H ). A subgraph H of G is called a spanning subgraph if V (H ) = V (G).

Let S be a nonempty subset of V (G). The induced subgraph by S , denoted

by G[S ], is a subgraph of G whose vertex-set is S and whose edge-set is the set of

those edges of G that have both end-vertices in S . The symbol G − S denotes the

induced subgraph G[V \ S ].

Let B be a nonempty subset of E (G), the edge-induced subgraph by B,

denoted by G[B], is a subgraph of G whose vertex-set is the set of end-vertices

of edges in B and whose edge-set is B. The symbol G − B denotes the spanning

subgraph G[E \ B] of G. Similarly, the graph obtained by adding a set of extra

edges F to G is denoted by G + F . Subgraphs of these various types are depicted

in Figure 1.10.x1

x2

x3x4

x5

e1

e2

e3

e4

e5

e6

e7

e8

G

x1

x2

x3x4

x5

e1

e2

e3

e4

e5

e6

e8

A spanning subgraph of G

x2

x4

x5

e4

e7

e8

G − {x1, x3}

x1

x2

x3x4

x5

e2

e3

e4 e6

e7

e8

G − {e1, e5}

x1

x2

x4

e1

G[{x1, x2, x4}]

x1

x2

x3x4

x5

e1

e3

e5

e8

G[{e1, e3, e5, e8}]

Figure 1.10: A graph and its various types of subgraphs

Operations: Let G1 and G2 be subgraphs of G. We say that G1 and G2 are

disjoint if they have no vertex in common, and edge-disjoint if they have no edge

in common.

The union G1∪G2 of G1 and G2 is the subgraph with vertex-set V (G1)∪V (G2)




and edge-set E (G1) ∪ E (G2). We write G1 + G2 for G1 ∪ G2 if G1 and G2 are

disjoint, and G1 ⊕G2 for G1 ∪G2 if G1 and G2 are edge-disjoint. If Gi ∼= H for each

i = 1, 2, · · · , n, then write nH for G1 + G2 + · · · + Gn.

The intersection G1∩G2 of G1 and G2 is defined similarly if V (G1)∩V (G2) = ∅.

These operations of graphs are depicted in Figure 1.11.

x1

x2 x3

∪

x2 x3

x4

=

x1

x2 x3

x4

x1

x2 x3

∩

x2 x3

x4

=

x2 x3x1

x2

x3 x4

x5

⊕

x1

x2

x3 x4

x5

=

x1

x2

x3 x4

x5

Figure 1.11: Union and intersection of graphs

An edge e of G is said to be contracted if it is deleted and its end-vertices are

identified; the resulting graph is denoted by G · e. This is illustrated in Figure 1.12.

x2

x1

x5

x4 x3

G

x1

x2

x3 = x5

x4

G · eFigure 1.12: A graph G · e by contracting the edge e of G

Example 1.4.1 Let G be a balanced digraph. Then d+G(X ) = d−

G(X ) for

any nonempty proper X ⊂ V (G).

Proof: Let H = G[X ]. Since G is balanced, d+G(x) = d−

G(x) for each x ∈ V (G).

By Theorem 1.1, we have that x∈X

d+H (x) =

x∈X

d−H (x). Thus,

d+G(X ) =

x∈X

d+G(x) −

x∈X

d+H (x) =

x∈X

d−G(x) −

x∈X

d−H (x) = d−

G(X )

as required.

Example 1.4.2 Let G be an undirected graph without loops. Then G

contains a bipartite spanning subgraph H such that dG(x) ≤ 2dH (x) for any x ∈

V (G). Hence ε(G) ≤ 2 ε(H ).



1.4. SUBGRAPHS AND OPERATIONS 17

Proof: Let H be a bipartite spanning subgraph of G with as many edges as

possible, and let {X, Y } be a bipartition. Arbitrarily choose x ∈ V (G), without loss

of generality, say x ∈ X . Let d = dG(x) − dH (x).

We claim that d ≤ dH (x). In fact, suppose to the contrary that d > dH (x). Let

X ′ = X \ {x} and Y ′ = Y ∪ {x}. Consider a bipartite spanning subgraph H ′ of G

with the bipartition {X ′, Y ′}. Then

ε(H ) ≥ ε(H ′) = ε(H ) + d − dH (x) > ε(H ),

a contradiction. Thus, dG(x) = d + dH (x) ≤ 2 dH (x). Summing up all vertices in G

yields that ε(G) ≤ 2 ε(H ) by Corollary 1.1.

Cartesian Product of Graphs: The cartesian product G1 × G2 of

two simple graphs G1 and G2 is a graph with the vertex-set V 1 × V 2, in which

there is an edge from a vertex x1x2 to another y1y2, where x1, y1 ∈ V (G1) and

x2, y2 ∈ V (G2), if and only if either x1 = y1 and (x2, y2) ∈ E (G2), or x2 = y2 and

(x1, y1) ∈ E (G1). See

Figure 1.8, for example, Q2 = K 2 × K 2, Q3 = K 2 × Q2 and Q4 = K 2 × Q3, in

general, Qn = K 2 × Qn−1.

Some simple properties are stated in the exercise 1.4.6. Particularly, the cartesian

product satisfies commutative and associative laws if we identify isomorphic

graphs. It is the two laws that can make us greatly simplify proofs of many propertiesof the cartesian products.

Let Gi = (V i, E i) be a graph for each i = 1, 2, · · · , n. By the commutative and

associative laws of the cartesian product, we may write G1 × G2 × · · · × Gn for the

cartesian product of G1, G2, · · · , Gn, where V (G1×G2×···×Gn) = V 1×V 2×···×V n.

Two vertices x1x2 · · ·xn and y1y2 · · · yn are linked by an edge in G1 ×G2 × · · · ×Gn

if and only if two vectors (x1, x2, · · · , xn) and (y1, y2, · · · , yn) differ exactly in one

coordinate, say the ith, and there is an edge (xi, yi) ∈ E (Gi).

Example 1.4.3 An important class of graphs, the well-known hypercube

Qn, defined in Example 1.2.1, can be defined in terms of the cartesian products,

that is,

Qn = K 2 × K 2 × · · · ×K 2 n

of n identical complete graph K 2, see Figure 1.8 for Q1, Q2, Q3 and Q4. The

hypercube is an important class of topological structures of interconnection networks,

some of whose properties will be further discussed in some sections in this book.




Line Graphs: The line graph of G, denoted by L(G), is a graph with

vertex-set E (G) in which there is an edge (a, b) if and only if there are vertices

x,y,z ∈ V (G) such that ψG(a) = (x, y) and ψG(b) = (y, z). This is illustrated in

Figure 1.13. Some simple properties of line graphs are stated in the exercise 1.4.4.

0 1

0001

10

11

B(2, 1) = K +

2

00

10

01

11

000001

010

011

100

101

110

111

B(2, 2) = L(B(2, 1))

000

100

001

010 101

110

011

111

B(2, 3) = L(B(2, 2))

Figure 1.13: Graphs and their line graphs

Assume that L(G) is the line graph of a graph G. If L(G) is non-empty and has

no isolated vertices, then its line graph L(L(G)) exists. For integers n ≥ 1,

Ln(G) = L(Ln−1(G)),

where L0(G) and L1(G) denote G and L(G), respectively, and Ln−

1(G) is assumed

to be non-empty and has no isolated vertex. The graph Ln(G) is called the nth

iterated line graph of a graph G.

Example 1.4.4 Two important classes of graphs, the well-knownn-dimensional

d-ary Kautz digraph K (d, n) and de Bruijn digraphs B(d, n) can be defined as

K (d, n) = Ln−1(K d+1) B(d, n) = Ln−1(K +d ),

where K +d (d ≥ 2) denotes a digraph obtained from a complete digraph K d by ap-

pending one loop at each vertex. The digraphs in Figure 1.13 are B(2, 1), B(2, 2)

and B(2, 3).

The original definitions of K (d, n) and B(d, n) will be given in Section 1.8,

Exercises: 1.3.5, 1.3.6

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Graph theory chapter 2