1 Introduction to Graph Theory


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1 CHAPTER 1 Introduction to Graph Theory INTRODUCTION It is no coincidence that graph theory has been independently discovered many times, since it may quite properly be regarded as an area of applied mathematics. The basic combinatorial nature of graph theory and a clue to its wide applicability are indicated in the words of Sylvester, The theory of ramification is one of pure colligation, for it takes no account of magnitude or position ; geometrical lines are used, but have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation. Indeed, the earliest recorded mention of the subject occurs in the works of Euler, and although the original problem he was considering might be regarded as a somewhat frivolous puzzle, it did arise from the physical world. Subsequent rediscoveries of graph theory by Kirchhoff and Cayley also had their roots in the physical world. Kirchhoff s investigations of electric networks led to his development of the basic concepts and theorems concerning trees in graphs, while Cayley considered trees arising from the enumeration of organic chemical isomers. Another puzzle approach to graphs was proposed by Hamilton. After this, the celebrated four colour conjecture came into prominence and has been notorious ever since. In the present century, there have already been a great many rediscoveries of graph theory which we can only mention most briefly in this chronological account. Euler ( ) became the father of graph theory as well as topology. Graph theory is considered to have begun in 1736 with the publication of Euler s solution of the Königsberg bridge problem. The graph theory is one of the few fields of mathematics with a definite birth date by ore. 1.1 WHAT IS A GRAPH? DEFINITION A graph G consists of a set of objects V = {v 1, v 2, v 3,...} called vertices (also called points or nodes) and other set E = {e 1, e 2, e 3,...} whose elements are called edges (also called lines or arcs). The set V(G) is called the vertex set of G and E(G) is the edge set. Usually the graph is denoted as G = (V, E) Let G be a graph and {u, v} an edge of G. Since {u, v} is 2element set, we may write {v, u} instead of {u, v}. It is often more convenient to represent this edge by uv or vu. If e = uv is an edge of a graph G, then we say that u and v are adjacent in G and that e joins u and v. (We may also say that each that of u and v is adjacent to or with the other). For example : A graph G is defined by the sets V(G) = {u, v, w, x, y, z} and E(G) = {uv, uw, wx, xy, xz}. 1
2 2 GRAPH THEORY WITH APPLICATIONS Now we have the following graph by considering these sets. Every graph has a diagram associated with it. The vertex u and an edge e are incident with each other as are v and e. If two distinct edges say e and f are incident with a common vertex, then they are adjacent edges. A graph with pvertices and qedges is called a (p, q) graph. The (1, 0) graph is called trivial graph. In the following figure the vertices a and b are adjacent but a and c are not. The edges x and y are adjacent but x and z are not. Although the edges x and z intersect in the diagram, their intersection is not a vertex of the graph. Examples : (1) Let V = {1, 2, 3, 4} and E = {{1, 2}, {1, 3}, {3, 2}. {4, 4}}. Then G(V, E) is a graph. (2) Let V = {1, 2, 3, 4} and E = {{1, 5}, {2, 3}}. Then G(V, E) is not a graph, as 5 is not in V. (3) A graph with 5vertices and 8edges is called a (5, 8) graph.
3 INTRODUCTION TO GRAPH THEORY DIRECTED AND UNDIRECTED GRAPHS Directed graph A directed graph or digraph G consists of a set V of vertices and a set E of edges such that e E is associated with an ordered pair of vertices. In other words, if each edge of the graph G has a direction then the graph is called directed graph. In the diagram of directed graph, each edge e = (u, v) is represented by an arrow or directed curve from initial point u of e to the terminal point v. Figure 1(a) is an example of a directed graph. Fig. 1(a). Directed graph. Suppose e = (u, v) is a directed edge in a digraph, then (i) u is called the initial vertex of e and v is the terminal vertex of e (ii) e is said to be incident from u and to be incident to v. (iii) u is adjacent to v, and v is adjacent from u Undirected graph An undirected graph G consists of set V of vertices and a set E of edges such that each edge e E is associated with an unordered pair of vertices. In other words, if each edge of the graph G has no direction then the graph is called undirected graph. Figure 1(b) is an example of an undirected graph. We can refer to an edge joining the vertex pair i and j as either (i, j) or (j, i). Figure 1(b). Undirected graph.
4 4 GRAPH THEORY WITH APPLICATIONS 1.3 BASIC TERMINOLOGIES Loop : An edge of a graph that joins a node to itself is called loop or self loop. i.e., a loop is an edge (v i, v j ) where v i = v f Multigraph In a multigraph no loops are allowed but more than one edge can join two vertices, these edges are called multiple edges or parallel edges and a graph is called multigraph. Two edges (v i, v j ) and (v f, v r ) are parallel edges if v i = v r and v j, v f. Fig. 2(a) Fig. 2(b) In Figure 1.2(a), there are two parallel edges associated with v 2 and v 3. In Figure 1.2(b), there are two parallel edges joining nodes v 1 and v 2 and v 2 and v Pseudo graph A graph in which loops and multiple edges are allowed, is called a pseudo graph. Fig. 3(a) Fig. 3(b) Simple graph A graph which has neither loops nor multiple edges. i.e., where each edge connects two distinct vertices and no two edges connect the same pair of vertices is called a simple graph. Figure 1.1(a) and (b) represents simple undirected and directed graph because the graphs do not contain loops and the edges are all distinct Finite and Infinite graphs A graph with finite number of vertices as well as a finite number of edges is called a finite graph. Otherwise, it is an infinite graph.
5 INTRODUCTION TO GRAPH THEORY DEGREE OF A VERTEX The number of edges incident on a vertex v i with selfloops counted twice (is called the degree of a vertex v i and is denoted by deg G (v i ) or deg v i or d(v i ). The degrees of vertices in the graph G and H are shown in Figure 4(a) and 4(b). Fig. 4(a) Fig. 4(b) In G as shown in Figure 4(a), deg G (v 2 ) = 2 = deg G (v 4 ) = deg G (v 1 ), deg G (v 3 ) = 3 and deg G (v 5 ) = 1 and In H as shown in Figure 4(b), deg H (v 2 ) = 5, deg H (v 4 ) = 3, deg H (v 3 ) = 5, deg H (v 1 ) = 4 and deg H (v 5 ) = 1. The degree of a vertex is some times also referred to as its valency. 1.5 ISOLATED AND PENDENT VERTICES Isolated vertex A vertex having no incident edge is called an isolated vertex. In other words, isolated vertices are those with zero degree Pendent or end vertex A vertex of degree one, is called a pendent vertex or an end vertex. In the above Figure, v 5 is a pendent vertex In degree and out degree In a graph G, the out degree of a vertex v i of G, denoted by out deg G (v i ) or number of edges beginning at v i and the in degree of v i, denoted by in deg G (v i ) or deg G + 1 deg G (v i ), is the (v i ), is the number of edges ending at v i. The sum of the in degree and out degree of a vertex is called the total degree of the vertex. A vertex with zero in degree is called a source and a vertex with zero out degree is called a sink. Since each edge has an initial vertex and terminal vertex. 1.6 THE HANDSHAKING THEOREM 1.1 If G = (v, E) be an undirected graph with e edges. i.e., Then v V deg ( v) = 2e G the sum of degrees of the vertices is an undirected graph is even.
6 6 GRAPH THEORY WITH APPLICATIONS Proof : Since the degree of a vertex is the number of edges incident with that vertex, the sum of the degree counts the total number of times an edge is incident with a vertex. Since every edge is incident with exactly two vertices, each edge gets counted twice, once at each end. Thus the sum of the degrees equal twice the number of edges. Note : This theorem applies even if multiple edges and loops are present. The above theorem holds this rule that if several people shake hands, the total number of hands shake must be even that is why the theorem is called handshaking theorem. Corollary : In a non directed graph, the total number of odd degree vertices is even. Proof : Let G = (V, E) a non directed graph. Let U denote the set of even degree vertices in G and W denote the set of odd degree vertices. Then deg G ( vi ) = G vi V vi U deg ( vi ) + deg G ( vi ) vi W 2e deg G ( v1) = deg G ( v1)...(1) v i U v i W Now deg G ( vi ) is also even vi W Therefore, from (1) deg G ( vi ) is even vi W The no. of odd vertices in G is even. Theorem 1.2. If V = {v 1, v 2,... v n } is the vertex set of a non directed graph G, then n i = 1 deg ( v ) = 2 E i If G is a directed graph, then n n + vi i = 1 i = 1 deg ( ) = deg ( v ) = E Proof : Since when the degrees are summed. Each edge contributes a count of one to the degree of each of the two vertices on which the edge is incident. Corollary (1) : In any non directed graph there is an even number of vertices of odd degree. Proof : Let W be the set of vertices of odd degree and let U be the set of vertices of even degree. Then deg ( v) = deg ( v) + deg ( v) = 2 E v V(G) v W v U i Certainly, deg ( v) is even, v U
7 INTRODUCTION TO GRAPH THEORY 7 Hence deg ( v) is even, v W Implying that W is even. Corollary (2) : If k = δ(g) is the minimum degree of all the vertices of a non directed graph G, then k V v V(G) deg ( v) = 2 E In particular, if G is a kregular graph, then k V = deg ( v) = 2 E. v V(G) Problem 1.1. Show that, in any gathering of six people, there are either three people who all know each other or three people none of whom knows either of the other two (six people at a party). Solution. To solve this problem, we draw a graph in which we represent each person by a vertex and join two vertices by a solid edge if the corresponding people know each other, and by a dotted edge if not. We must show that there is always a solid triangle or a dotted triangle. Let v be any vertex. Then there must be exactly five edges incident with v, either solid or dashed, and so at least three of these edges must be of the same type. Let us assume that there are three solid edges (see figure 5) ; the case of atleast three dashed edges is similar. Fig. 5. If the people corresponding to the vertices w and x know each other, then v, w and x form a solid triangle, as required. Similarly, if the people corresponding to the vertices w and y, or to the vertices x and y, know each other, then we again obtain a solid triangle. These three cases are shown in Figure (6).
8 8 GRAPH THEORY WITH APPLICATIONS Fig. 6. Finally, if no two of the people corresponding to the vertices w, x and y know each other, then w, x and y from a dotted triangle, as required (see figure (7). Fig. 7. Problem 1.2. Place the letters A, B, C, D, E, F, G, H into the eight circles in Figure (8), in such a way that no letter is adjacent to a letter that is next to it in the alphabet. Fig. 8. Solution. First note that trying all the possibilities is not a practical proposition, as there are 8! = ways of placing eight letters into eight circles. Note that (i) the easiest letters to place are A and H, because each has only one letter to which it cannot be adjacent, namely, B and G, respectively. (ii) the hardest circles to fill are those in the middle, as each is adjacent to six others. This suggests that we place A and H in the middle circles. If we place A to the left of H, then the only possible positions for B and G are shown in Figure (9). Fig. 9.
9 INTRODUCTION TO GRAPH THEORY 9 The letter C must now be placed on the lefthand side of the diagram, and F must be placed on the righthand side. It is then a simple matter to place the remaining letters, as shown in Figure (10). Fig. 10. Problem 1.3. Determine the number of edges in a graph with 6 vertices, 2 of degree 4 and 4 of degree 2. Draw two such graphs. Solution. Suppose the graph with 6 vertices has e number of edges. Therefore by Handshaking lemma 6 i = 1 deg ( v ) = 2e i d(v 1 ) + d(v 2 ) + d(v 3 ) + d(v 4 ) + d(v 5 ) + d(v 6 ) = 2e Now, given 2 vertices are of degree 4 and 4 vertices are of degree 2. Hence the above equation, (4 + 4) + ( ) = 2e 16 = 2e e = 8. Hence the number of edges in a graph with 6 vertices with given condition is 8. Two such graphs are shown below in Figure (11). Fig. 11. Problem 1.4. How many vertices are needed to construct a graph with 6 edges in which each vertex is of degree 2. Solution. Suppose these are P vertices in the graph with 6 degree. Also given the degree of each vertex is 2. By handshaking lemma, P i = 1 deg ( v ) = 2q= 2 6 i d(v 1 ) + d(v 2 ) d(v n ) = = 12 2P = 12 P = 6 vertices are needed.
10 10 GRAPH THEORY WITH APPLICATIONS Problem 1.5. It is possible to construct a graph with 12 vertices such that 2 of the vertices have degree 3 and the remaining vertices have degree 4. Solution. Suppose it is possible to construct a graph with 12 vertices out of which 2 of them are having degree 3 and remaining vertices are having degree 4. Hence by handshaking lemma, 12 dv ( i ) = 2e where e is the number of edges i = 1 According to given conditions (2 3) + (10 4) = 2e = 2e 2e = 46 e = 23 It is possible to construct a graph with 23 edges and 12 vertices which satisfy given conditions. Problem 1.6. It is possible to draw a simple graph with 4 vertices and 7 edges? Justify. Solution. In a simple graph with Pvertices, the maximum number of edges will be P(P 1) Hence a simple graph with 4 vertices will have at most = 6 edges. 2 Therefore, the simple graph with 4 vertices cannot have 7 edges. Hence such a graph does not exist. Problem 1.7. Show that the maximum degree of any vertex in a simple graph with P vertices is (P 1). Solution. Let G be a simple graph with Pvertices. Consider any vertex v of G. Since the graph is simple (i.e., without self loops and parallel edges), the vertex v can be adjacent to atmost remaining (P 1) vertices. Hence the maximum degree of any vertex in a simple graph with P vertices is (P 1). Problem 1.8. Write down the vertex set and edge set of the following graphs shown in Figure 12(a) and 12(b). Fig. 12.
(a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from
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