1.3 Vertex Degrees. Vertex Degree for Undirected Graphs: Let G be an undirected. Vertex Degree for Digraphs: Let D be a digraph and y V (D).

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1 1.3. VERTEX DEGREES Vertex Degrees Vertex Degree for Undirected Graphs: Let G be an undirected graph and x V (G). The degree d G (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, d G ( ) = d G ( ) = 4, d G ( ) = d G ( ) = 3. e e 2 1 a 1 a 2 e 7 e 3 a 7 a 4 a 3 e 6 a 5 e 5 (a) Figure 1.9: (a) an undirected graph G a 6 (b) (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 d G (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{d G (x) : x V (G)}. The minimum degree of G: δ(g) = min{d G (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 (y 1) = 2, d + D (y 2) = 1, d + D (y 3) = 1, d + D (y 4) = 3; d D (y 1) = 2, d D (y 2) = 2, d D (y 3) = 3, d D (y 4) = 0, A vertex y is called to be balanced if d + D (y) = d D (y), and D is called to be

2 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)}, δ(d) = min {δ + (D), δ (D)} and 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 bipartite {X, Y }. It is easy to see that the relationship between degree of vertices and the number of edges of G is as follows. d G (x) = ε(g) = d G (y). (1.3) x X y Y As a result, we have 2 ε(g) = d G (x). (1.4) x V (G) 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 d G (x ) = d + D (x), d G(x ) = d D (x), x V (D). By the equality (1.3), we have that d + D (x) = d G (x ) = ε(g) = d G (x ) = d D (x). x V x X x Y x V Since ε(d) = ε(g) by (1.2), the theorem follows. Corollary 1.1 For any undirected graph G, 2ε(G) = x V d G (x)

3 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 By Theorem 1.1, we have that d G (x) = d + D (x) = d D (x), x V. d G (x) = d + D (x) = x V x V 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. Then d G (x) + d G (x) = d G (x) = 2ε(G). x V o x V e x V Since d G (x) is even, it follows that d G (x) is also even. Since d G (x) is odd x V e x V o 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 µ D (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 µ D (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 = {y 1, y 2 }, then E + D (S) = {a 3}, d + D (S) = 1, N+ D (S) = {y 3}, E D (S) = {a 4, a 7 }, d D (S) = 2, N D (S) = {y 3, y 4 }. 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, d G (S) = E G (S). Example Prove that ε(g) 1 4 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 [d G (x) 1] + [d G (y) 1] v 2,

4 14 Basic Concepts of Graphs that is, Then summing over all edges in G yields d G (x) + d G (y) v. x V d 2 G (x) v ε. By Cauchy s inequality and Corollary 1.1, we have that v ε x V d 2 G(x) 1 v ( ) 2 d G (x) = 4 v ε2, x V that is, ε(g) 1 4 v2. Example 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) : d G (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 d G c(x) 1 2 (v 1) since G = G c. Then there must exist y x V (G) with d G (y x ) = d G c(x). Note that d G (y x ) = d G c(x) = (v 1) d G c(x) 1 2 (v 1). (1.5) Thus, y x x from (1.5) and y x V. Furthermore, y x y z if x, z V and x z. This fact implies that the vertices in V occur in pairs, which shows that V is even.

5 1.4. SUBGRAPHS AND OPERATIONS 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), ψ G ) is a graph. A graph H = (V (H), E(H), ψ H ) 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 ψ H is the restriction of ψ G 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 e 5 e 7 e 1 e 5 e 1 e 7 e 6 e 2 e 6 e 2 e 3 e 3 G A spanning subgraph of G G {, } e 7 e 1 e 5 e 1 e6 e 2 e 3 e 4 G {e 1, e 5} G[{,, }] G[{e 1, e 3, e 5, }] Figure 1.10: A graph and its various types of subgraphs Operations: Let G 1 and G 2 be subgraphs of G. We say that G 1 and G 2 are disjoint if they have no vertex in common, and edge-disjoint if they have no edge in common. The union G 1 G 2 of G 1 and G 2 is the subgraph with vertex-set V (G 1 ) V (G 2 )

6 16 Basic Concepts of Graphs and edge-set E(G 1 ) E(G 2 ). We write G 1 + G 2 for G 1 G 2 if G 1 and G 2 are disjoint, and G 1 G 2 for G 1 G 2 if G 1 and G 2 are edge-disjoint. If G i = H for each i = 1, 2,,n, then write nh for G 1 + G G n. The intersection G 1 G 2 of G 1 and G 2 is defined similarly if V (G 1 ) V (G 2 ). These operations of graphs are depicted in Figure = = = 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 = G G e Figure 1.12: A graph G e by contracting the edge e of G Example 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 d + H (x) = d H (x). Thus, d + G (X) = x X x X d + G (x) x X x X d + H (x) = x X d G (x) x X d H (x) = d G (X) as required. Example Let G be an undirected graph without loops. Then G contains a bipartite spanning subgraph H such that d G (x) 2d H (x) for any x V (G). Hence ε(g) 2 ε(h).

7 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 = d G (x) d H (x). We claim that d d H (x). In fact, suppose to the contrary that d > d H (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 d H (x) > ε(h), a contradiction. Thus, d G (x) = d + d H (x) 2 d H (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 G 1 G 2 of two simple graphs G 1 and G 2 is a graph with the vertex-set V 1 V 2, in which there is an edge from a vertex to another y 1 y 2, where, y 1 V (G 1 ) and, y 2 V (G 2 ), if and only if either = y 1 and (, y 2 ) E(G 2 ), or = y 2 and (, y 1 ) E(G 1 ). See Figure 1.8, for example, Q 2 = K 2 K 2, Q 3 = K 2 Q 2 and Q 4 = K 2 Q 3, in general, Q n = K 2 Q n 1. Some simple properties are stated in the exercise 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 properties of the cartesian products. Let G i = (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 G 1 G 2 G n for the cartesian product of G 1, G 2,,G n, where V (G 1 G 2 G n ) = V 1 V 2 V n. Two vertices x n and y 1 y 2 y n are linked by an edge in G 1 G 2 G n if and only if two vectors (,,,x n ) and (y 1, y 2,,y n ) differ exactly in one coordinate, say the ith, and there is an edge (x i, y i ) E(G i ). Example An important class of graphs, the well-known hypercube Q n, defined in Example 1.2.1, can be defined in terms of the cartesian products, that is, Q n = K 2 K 2 K 2 }{{} n of n identical complete graph K 2, see Figure 1.8 for Q 1, Q 2, Q 3 and Q 4. 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.

8 18 Basic Concepts of Graphs 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 ψ G (a) = (x, y) and ψ G (b) = (y, z). This is illustrated in Figure Some simple properties of line graphs are stated in the exercise B(2, 1) = K B(2, 2) = L(B(2, 1)) B(2, 3) = L(B(2, 2)) 110 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, L n (G) = L(L n 1 (G)), where L 0 (G) and L 1 (G) denote G and L(G), respectively, and L n 1 (G) is assumed to be non-empty and has no isolated vertex. The graph L n (G) is called the nth iterated line graph of a graph G. Example Two important classes of graphs, the well-known n-dimensional d-ary Kautz digraph K(d, n) and de Bruijn digraphs B(d, n) can be defined as K(d, n) = L n 1 (K d+1 ) B(d, n) = L n 1 (K + d ), where K + d (d 2) denotes a digraph obtained from a complete digraph K d by appending 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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