University of Zimbabwe HMTH215 Graph Theory

Honours Graph Theory Department: Mathematics

Author: P. Mafuta

April 6, 2016

Chapter 1 Introduction: Basic Graph Theory This course serves to answer many questions arise in ordinary life and science in general. Just to mention but a few, it is used to select the best chemical in chemistry, designing networks, minimize cost, manage corruption and solve data communication problems.

Definition 1.0.1. A graph G = (V, E) consists of a finite non-empty set V (G) of elements called vertices and a (possibly empty) set E(G) of elements called edges. V (G) is called the vertex set of G and E(G) is called an edge set of G. NB A graph is not drawn uniquely, what matters is that the same vertices are joined together by the same edges. Definition 1.0.2. If two vertices of a graph are joined by two or more edges, then this graph has multiple edges. An edge joining a vertex to itself is called a loop. Definition 1.0.3. A graph with one vertex is called a trivial graph, otherwise the graph is nontrivial. Definition 1.0.4. A simple graph is a graph that has no loops or multiple edges. Definition 1.0.5. Two vertices of the same edge are said to be incident with the edge and two edges sharing a common vertex are said to be incident with the vertex. Two vertices incident with an edge are said to be adjacent.

1.0.1

Adjacent and Incident matrices of a graph

Definition 1.0.6. Let G = (V, E) be a graph such that V (G) = {v1 , v2 , v3 , . . . , vn } and E(G) = {e1 , e2 , e3 , . . . , em }. We define the adjacent matrix, say A, of G to be the n × n matrix where the ij th entry is the number of edges joining vi to vj . We define the incident matrix, say M , of G to be the n × m matrix where the ij th entry is 1 if vertex vi is incident to the edge ej and 0 otherwise. 1

Definition 1.0.7. Let G be a graph. Then a graph H = (V (H), E(H)) is a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). Thus a graph is a subgraph of itself. Definition 1.0.8. Let G = (V, E) be a graph. The degree or valency of a vertex v ∈ V (G) is the number of edges incident with v, where a loop contributes two units to the degree. We denote the degree of a vertex v by degG (v). A vertex of degree zero is called an isolated vertex. Definition 1.0.9. Let G be a graph. The number of vertices in V (G) is called the order of graph G and we denote it by n. The number of edges in E(G) is called the size of G and we denote it by m. Lemma 1.0.1. Handshaking lemma: In any graph, the sum of all vertex degree is equal to twice the number of edges, i.e., X degG (v) = 2m v∈V

Proof. We ignore vertices of degree zero since they contribute nothing to the sum. We notice that each edge has exactly two vertices, i.e., a vertex at each end. So each edge contributes 2 units to the sum of the degrees. Hence the result follows. Corollary 1.0.1. In any graph G

(i) The sum of all vertex degrees is an even number. (ii) The number of vertices of odd degree is even

Proof. Item (i) follows directly from the lemma above. To prove item (ii), suppose contrary to the claim that the number of vertices of odd degree is odd. Then the sum of all vertex degrees of vertices of odd degree is odd. Further, we note that the sum of all vertex degrees of all vertices of even degree is even. This implies that the sum of all vertex degree is odd, a contradiction to the handshaking lemma. Hence our supposition is false and the result follows. Definition 1.0.10. A sequence of degrees of vertices in a graph given in descending order is called the degree sequence. A sequence of non-negative integers is called graphic if the exists a graph whose degree sequence is precisely that sequence.

NB From the lemma above it is clear that if the sum of the terms of a sequence is odd then the sequence is not graphic. However, its not always true to say that if the sum of the terms of a sequence is even then the sequence is graphic. This calls for the follow algorithm to determine whether or not a given sequence is graphic. Theorem 1.0.1. Havel Hakimi: Consider the following two sequences and assume (i) is in descending order.

(i) s, t1 , t2 , . . . , ts , d1 , d2 , . . . , dn 2

(ii) t1 − 1, t2 − 1, . . . , ts − 1, d1 , . . . , dn . Then (i) is graphic iff (ii) is graphic. Proof. Ex

For instance, the sequence 6, 6, 6, 6, 4, 3, 3, 0 can be reduced using the above theorem to 3, 1, 0, 0, 0 which is not graphic, since there is no graph with one vertex of degree 3, one vertex of degree 1 and three vertices of degree 0. Hence the original sequence also is not graphic.

1.1

Types of Graphs

Definition 1.1.1. Let G = (V (G), E(G)) and H = (V (H), E(H)) be graphs. Then

(a) G and H are identical iff V (G) = V (H) and E(G) = E(H). (b) G and H of order n are said to be isomorphic if the vertices of G and H can be labelled with numbers 1 to n such that whenever vertex i is adjacent to vertex j in G, then i is adjacent to vertex j in H and conversely. Such a labeling is the same as one to one correspondence between V (G) and V (H) that preserves adjacency. Thus G and H are isomorphic if there is a bijection θ : V (G) −→ V (H) such that uv ∈ E(G) iff θ(u)θ(v) ∈ E(H).

Remark: Saying G and H are isomorphic amounts to saying that G can be relabelled to get another graph, say G0 , identical to H. In general, for large graphs it is difficult to tell whether or not two graphs are isomorphic. However, if two graphs are isomorphic, then their degree sequence must be the same. Thus if we are asked to tell whether given graphs are isomorphic, the first thing we have to do is to check whether they have the same degree sequence. Then if the degree sequence is the same, we have to do some more checking, since other graphs have the same degree sequence but are not isomorphic. Definition 1.1.2. A graph G is called a regular graph if all the vertices are of the same degree, i.e., degG (v) = k, ∀v ∈ V (G) for some k. We then call G a k−regular graph. A 3−regular graph is called a cubic graph, for instance the famous Petersen graph on the cover page, which we shall discuss later. Definition 1.1.3. A complete graph, which we denote by Kn , is a simple graph of order n such every vertex is adjacent to every other vertex, i.e., ∀u, v ∈ V (Kn ), u and v are adjacent. For n(n − 1) edges. In addition, instance, a trivial graph is a complete K1 graph. In general, Kn has 2 Kn has n subgraphs isomorphic to Kn−1 . Definition 1.1.4. A graph G such that E(G) = ∅ is called a null graph. Thus in a null graph, every vertex is an isolated vertex. 3

Definition 1.1.5. A graph G such that V (G) can be split into two disjoint sets V1 and V2 in such that every edge of G vertex of V1 to a vertex of V2 is called a bipartite graph. If G is a simple graph then G is called a complete bipartite graph. We denote a complete bipartite graph by Km,n where in this case |V1 | = m and |V2 | = n.

A K1,n is called a star graph. A star K1,3 is called a claw. Definition 1.1.6. Let G = (V, E) be any graph, and let S ⊂ V be any subset of vertices of G. Then the induced subgraph, say G[S] is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S. A claw-free graph is a graph that does not have a claw as an induced subgraph.

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Chapter 2 Connectedness in graphs Definition 2.0.7. A walk, W say, in a graph G is an alternating sequence: v0 , e1 , v1 , e2 , v2 , e3 , . . . , vr−1 , er , vr of vertices and edges such that ei = vi−1 vi for i = 1, 2, 3, . . . , r and r is called the length of the walk. That is, the number of edges involved is its length. Remark. (a) In any walk, the last vertex on an edge is the first vertex of the next edge. (b) A vertex maybe repeating in a walk, i.e., one can have a walk of the form u, uv.v, vw, w, wx, x, xv, v, vz, z etc. (c) An edge may be repeated in a walk. Definition 2.0.8. If all the edges of a walk are different, then the walk is called a trail. If all the vertices of a walk are different, then it is called a path. We denote by Puv a path from a vertex u to v, also known as a u − v path. Definition 2.0.9. Definition 2.0.10. In a walk W , if the initial vertex v0 and the final vertex vr are the same, i.e., v0 = vr , then the walk is called a closed walk. Further, if W is a closed walk with all vertices different except v0 and r ≥ 3, then W is called a cycle of length r or simply an r−cycle. We denote by Cn a cycle with n vertices. Definition 2.0.11. Let G1 = (V (G1 ), E(G1 )) and G2 = (V (G2 ), E(G2 )) be any two graphs. Then the union G1 ∪ G2 of G1 and G2 is the graph with vertex set V (G1 ) ∪ V (G2 ) and edge set E(G1 ) ∪ E(G2 ). NB We can remove an edge in a graph and/or any vertex (and the edge incident to it) to obtain another graph. If we remove the vertex v from G, we denote the resulting graph by G − v. If we 5

delete a set of vertices V 0 , we denote the resulting graph by G − V 0 (G − e or G − E 0 for edges). We call G − V 0 and G − E 0 vertex deleted subgraph and edge deleted subgraph, respectively. Also, we can remove the edge e = vw and identify v and w so that the resulting vertex v = w is incident to all edges that were originally incident with v or w. We denote the resulting graph by G \ {e} and call the deletion of e from G to obtain G \ {e} an edge contraction. A contraction of G is the graph obtained from G after a succession of edge contractions. Definition 2.0.12. A graph G is connected if the exists a u − v path for all u, v ∈ V (G), otherwise it is disconnected. The maximal connected subgraphs that make up a disconnected graph are called components of G.

A connected regular graph where each vertex is of degree two is called a circuit or cycle Cn graph denoted Cn . Theorem 2.0.1. Let G be a connected graph of order n and size m. Then n ≤ m + 1.

Proof. We prove by induction on the number of edges m. If m = 1 or m = 2 then the theorem holds. So assume that the theorem holds for each graph with fewer than k edges, i.e., for m < k. Let G be a graph with k edges and n vertices. We consider two cases. Case 2.0.1. Assume first that G contains a cycle. Then by removing one edge from the cycle, we obtain the graph H say, still connected and has k − 1 edges. The number of vertices of H is still n and by inductive hypothesis n ≤ (k − 1) + 1. So n ≤ k and certainly n ≤ k + 1. Hence we are done by principle of mathematical induction. Case 2.0.2. Now assume that G has no cycle. Let Pab be a longest path in G. Then degG (a) = degG (b) = 1, since a and b are end vertices of the path. Remove the vertex a and the edge incident to it. This yeilds a graph, say H, which is connected and has n − 1 vertices and k − 1 edges. So by inductive hypothesis n − 1 ≤ (k − 1) + 1, i.e., n ≤ k + 1. Thus we are done by principle of mathematical induction.

Definition 2.0.13. Two path in a graph are called edge disjoint if they have no edge in common. Similarly, they are called vertex disjoint if they have no vertex in common. Hence vertex disjoint path are edge disjoint but the converse is not necessarily true.

Some important definitions Definition 2.0.14. A cycle of length 3 is called a triangle. The graph that has no subgraph isomorphic to a triangle is called a triangle−free graph. A cycle of length 4 is denoted by C4 . A graph is called C4 −free if does not contain a cycle of length 4. The length of a shortest cycle in a graph is called girth and is denoted by g(G). In contrast, the length a longest cycle in G is called the circumference, denoted c(G).

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Definition 2.0.15. The distance, d(u, v), between any two vertices u and v in G is the length of a shortest path joining u and v. The eccentricity, ecc(v) of a vertex v ∈ V (G) is the distance from v to a vertex furthest from it in G. The maximum eccentricity is called the diameter and the minimum eccentricity is called the radius. We denote the diameter and the radius respectively by diam(G) and rad(G). Definition 2.0.16. A leaf or end vertex is a vertex v of graph G such that degG (v) = 1. A vertex of degree at least two is called an internal or interior vertex. Definition 2.0.17. A subgraph H of G is called a spanning subgraph if V (H) = V (G). Thus a spanning cycle of G is a cycle that contains all vertices of G. Similarly, a spanning path is a path that contains all vertices of G. Definition 2.0.18. The smallest of all the degrees of vertices in a graph G is called the minimum degree and the largest of all the degrees is called the maximum degree. We denote by δ(G) and bigtriangleup(G) the minimum degree and maximum degree, respectively. Definition 2.0.19. The length of a walk, W , is called even(odd) if there are even(odd) number of edges in the walk. Lemma 2.0.1. Let G be a bipartite graph. Then every cycle in G is of even length. Let V1 (G) and V2 (G) be two disjoint sets that forms V (G). Let v1 , v2 , v3 , . . . , vk , v1 be a cycle in G. Then it suffices to show that k is even. Assume without loss of generality that v1 ∈ V1 . Since G is a bipartite graph, v2 ∈ V2 , v3 ∈ V1 , v4 ∈ V2 , v5 ∈ V1 and so on. Thus vk must lie in V2 , since v1 ∈ V1 and vk is adjacent to v1 . Hence k must be even and the result follows. Theorem 2.0.2. Let G be a simple graph of order n. If G has k components, then its size m 1 satisfies n − k ≤ m ≤ (n − k)(n − k + 1). 2 Proof. Ex.

Connected and disconnected graphs Definition 2.0.20. A disconnecting set of a connected graph is a set of edges of G whose removal disconnects G. A cutset is any disconnecting set of G no proper subset of which is a disconnecting set. A bridge is any cutset that contains one edge, i.e., it is an edge whose removal disconnects G. Definition 2.0.21. Let G be a graph. The edge-connectivity, λ(G), of G is the size of the smallest cutset in G, i.e., the minimum number of edges whose removal disconnects G. If λ(G) ≥ k, then we say G is k−edge connected. A separating set of a connected graph G, is a set of vertices of G which when deleted disconnects G. A separating set with only one vertex is called a cut-vertex or an articulation-vertex, i.e., a cut-vertex is a vertex whose removal disconnects G. The minimum number of vertices whose deletion disconnects G is called the vertex-connectivity, denoted by κ(G). That is, κ(G) is the size of the smallest separating set. G is k−connected if κ(G) ≥ k. A vertex-cutset of a graph G is a separating set, S say, such that no proper subset of it disconnects G. 7

Theorem 2.0.3. Let G be a connected graph. Then κ(G) ≤ λ(G) ≤ δ(G).

Proof. Since δ(G) is the minimum degree, there exist at least one vertex of degree δ. Let v ∈ V (G) be a vertex of degree δ and u1 , u2 , u3 , . . . , uδ ∈ N (v). Then by deleting the edges vui , i = 1, 2, 3, . . . , δ, we disconnect the graph. Thus the minimum number of edges required to disconnect the graph can not exceed δ. That is λ(G) ≤ δ(G). Further, since λ(G) is the minimum number of edges whose removal disconnects G, there is a set of λ(G) edges whose removal disconnects the graph into at least two components. It follows that, we can also disconnect G by removing one suitably chosen vertex from each of these λ(G) edges. Hence the minimum number of vertices whose deletion disconnects G cannot exceeds the edge connectivity λ(G). So κ(G) ≤ λ(G) and the result follows. Definition 2.0.22. A connected graph with no cut vertices is called a block.

NB Every graph is a union of blocks. Futher a block with at least 3 vertices is 2−connected. Definition 2.0.23. A family of paths in a graph G is said to be internally disjoint if no vertex of G is an internal vertex of more than one path of the family. Theorem 2.0.4. Let G be a graph of order n ≥ 3. Then G is 2−connected iff any two vertices of G are connected by at least two internally disjoint paths.

Proof. Ex. Corollary 2.0.1. If G is 2−connected, then any two vertices of G lie on a common cycle. Definition 2.0.24. An edge of a graph G is said to be subdivided when it is deleted and replaced by a path of length two connecting its ends. The internal vertex of this path is thus a new vertex. Corollary 2.0.2. Let G be a block with order n ≥ 3. Then any two vertices lie on a common cycle. Theorem 2.0.5. Menger’s Theorem: Edge form. Let G be a connected graph, and u, v ∈ V (G) be any two vertices in G. Then the maximum number of edge disjoint paths from u to v is equal to the minimum number of edges separating u from v.

Proof. Ex. Corollary 2.0.3. A connected graph is k−connected if and only if any two vertices of G are connected by at least k−edge disjoint paths. Theorem 2.0.6. Menger’s Theorem: Vertex form. Let G be a connected graph, and u, v ∈ V (G) be any two non-adjacent vertices in G. Then the maximum number of vertex-disjoint paths from u to v is equal to the minimum number of vertices separating u from v.

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2.1

Trees and forest

Definition 2.1.1. A tree is a connected graph without cycles. A collection of trees is called a forest. A tree T is called a spanning tree of a graph G if it is a spanning subgraph of G. A graph with no cycles is called an acyclic graph. Definition 2.1.2. The leaf number, denoted by L(G), of a graph G is the maximum number of leaf vertices contained in a spanning tree of G. Its applications are legion in network designs and in solving data communication problems. Theorem 2.1.1. Every tree has at least two end vertices.

Proof. Ex. Theorem 2.1.2. Every connected graph G contains a spanning tree. Theorem 2.1.3. Let T be a graph with order n and size m. Then the following statements are equivalent:

1. T is a tree. 2. T is acyclic and n = m + 1. 3. T is connected and n = m + 1. 4. T is connected and T − e is disconnected ∀e ∈ E(T ). That is, every edge is a bridge. 5. ∀x, y ∈ V (T ), ∃! P (x, y), an x − y path in T . 6. T has no cycles but the addition of one new edge creates exactly one cycle.

Proof. (a) =⇒ (b). Assume that (a) holds. Then T is a tree and by definition it is a connected graph without cycles. Hence T is acyclic. So it is enough to show that n = m + 1. We prove this by induction on the number of edges m. If T = K1 then n = 1 and m = 0 and the result holds. So we assume that E(T ) 6= ∅ and that the result is true for every graph with m − 1 edges. Since T is a tree, it has at least two end vertices. Let x be a leaf vertex of T . Then degT (x) = 1. Let Tx = T − x be a graph obtained from T by deleting the vertex x. Then Tx has order n − 1 and size m − 1. So by inductive assumption, |V (Tx )| = |E(Tx )|+1. Thus n−1 = (m−1)+1 and so n = m+1 as desired. (b) =⇒ (c). Assume that (b) is true. Then T is acyclic and n = m + 1. It suffices to show that T is connected. Assume contrary to the claim that T is disconnected into k components. Then k ≥ 2, otherwise T is connected. Let T1 , T2 , T3 , . . . , Tk be the components of T where each Ti has order ni . Now for each i, Ti has no cycles and has ni − 1 edges. This implies that T (n1 − 1) + (n2 − 1) + . . . + nk − 1 edges, i.e., T has n − k edges, where k ≥ 2. This contradicts the fact that T has n − 1 edges. Therefore T must be connected. (c) =⇒ (d). Assume that (c) holds. Then T is connected and n = m + 1. It is adequate to 9

show that every edge of T is a bridge. Assume that T has an edge which is not a bridge. Then the removal of that edge yields a connected graph with n − 2 edges and n vertices, a contradiction. Hence T − e is disconnected ∀e ∈ E(T ). (d) =⇒ (e). Assume that (d) is true and suppose contrary to the claim that for some x, y ∈ V (T ) that there exists at least two paths from x to y. Then T has at least one cycle and the deletion of one edge on such a cycle does not disconnect the graph. This is a contradiction to the fact that every edge of T is a bridge. Hence (e) holds also. (e) =⇒ (f ). If (e) holds then clearly, T has no cycles otherwise there exists at least two paths from x to y for some x, y ∈ V (T ), which is not allowed. Further, from (e), we conclude that T is connected, hence the addition of a new edge creates a cycle in T . To show uniqueness, we suppose on contrary that the addition of one creates at least two cycles. Then it implies that some vertices of T are connected by more than one path, contradicting (e). Hence (e) implies (f ). (f ) =⇒ (a). Following (f ), we notice that T has no cycles. So it is enough to show that T is connected for it to be a tree. Suppose on contrary that T is disconnected. Then by adding an edge from one component of T to the other, we see that no cycle is created. This contradicts (f ). Hence T must be connected. This completes the proof.

2.2

Eulerian and Hamiltonian graphs

Definition 2.2.1. A graph G is Eulerian if contains a closed trail that includes every edge of edge. G is called semi-Eulerian if it contains a trail that includes edge. Thus every Eulerian graph is semi-Eulerian. Definition 2.2.2. A graph G is said to be Hamiltonian if it contains a spanning cycle. G is said to be traceable if it contains a spanning path. Definition 2.2.3. Definition 2.2.4.

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1 “Everybody is a genius. But if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.”— Albert Einstein

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