Bachelor Degree in Informatics Engineering Facultat d’Inform`aticade Barcelona Mathematics 1 Part I: Graph Theory Exercises and problems February 2019 Departament de Matem`atiques Universitat Polit`ecnicade Catalunya The problems of this collection were initially gathered by Anna de Mier and Montserrat Mau- reso. Many of them were taken from the problem sets of several courses taught over the years by the members of the Departament de Matem`aticaAplicada 2. Other exercises came from the bibliography of the course or from other texts, and some of them were new. Since Mathematics 1 was first taught in 2010 several problems have been modified or rewritten by the professors involved in the teaching of the course. We would like to acknowledge the assistance of the scholar Gabriel Bernardino in the writing of the solutions. Translation by Anna de Mier and the scholar Bernat Coma. This problem list has been revised during the academic year 2018/2019. Contents 1 Graphs: basic concepts 1 1.1 Types of graphs. Subgraphs. Operations with graphs. :::::::::::::: 1 1.2 Exercises :::::::::::::::::::::::::::::::::::::::::: 3 2 Walks, connectivity and distance 7 3 Eulerian and Hamiltonian graphs 10 4 Trees 12 Review exercises 15 1 Graphs: basic concepts 1.1 Types of graphs. Subgraphs. Operations with graphs. The following are some important families of graphs that we will use often. Let n be a positive integer and V = fx1; x2; : : : ; xng. The null graph of order n, denoted by Nn, is the graph of order n and size 0. The graph N1 is called the trivial graph. The complete graph of order n, denoted by Kn, is the graph of order n that has all possible edges. We observe that K1 is a trivial graph too. The path graph of order n, denoted by Pn = (V; E), is the graph that has as a set of edges E = fx1x2; x2x3; : : : ; xn−1xng. The cycle graph of order n ≥ 3, denoted by Cn = (V; E), is the graph that has as a set of edges E = fx1x2; x2x3; : : : ; xn−1xn; xnx1g. The wheel graph of order n ≥ 4, denoted by Wn = (V; E), is the graph that has as a set of edges E = fx1x2; x2x3; : : : ; xn−1x1g [ fxnx1; xnx2; : : : ; xnxn−1g. Let r and s be positive integers. A graph is r-regular if all vertices have degree r. A graph G = (V; E) is bipartite if there are two non-empty subsets V1 and V2 such that V = V1 [ V2, V1 \ V2 = ; and, for every edge uv 2 E, we have u 2 V1 and v 2 V2, or vice versa. That is, there are no edges uv with u; v 2 V1 or u; v 2 V2. The sets V1 and V2 are called the stable parts of G. If every vertex from V1 is adjacent to every vertex of V2, we say that the graph is complete bipartite and we denote it by Kr;s, where jV1j = r and jV2j = s. The graph K1;s is called a star graph. 2 Chapter 1. Graphs: basic concepts Subgraphs Let G = (V; E) be a graph. The graph G0 = (V 0;E0) is a subgraph of G if V 0 ⊆ V and E0 ⊂ E. If V 0 = V , it is called a spanning subgraph of G. Let S ⊆ V , S 6= ;. The graph G[S] = (S; E0) with E0 = fuv 2 E : u; v 2 Sg is called the subgraph induced (or spanned) by the set of vertices S . Graphs derived from a graph Consider a graph G = (V; E). The complement of G, denoted by Gc, is the graph with set of vertices V and set of edges Ec = fuvj uv 62 Eg. A graph isomorphic to its complement is called self-complementary. Let S ⊂ V . The graph obtained by deleting the vertices from S, denoted by G − S, is the graph having as vertices those of V n S and as edges those of G that are not incident to any vertex from S. In the case that S = fvg, we denote it G − v. Let S ⊂ E. The graph obtained by deleting the edges from S, denoted by G − S, is the graph obtained from G by removing all the edges from S. That is, G − S = (V; E n S). If S = feg, we write G − e. Let u; v be vertices from G that are not adjacent. The graph obtained by adding the edge uv is the graph G + uv = (V; E [ fuvg). Operations with graphs Consider two graphs G1 = (V1;E1) and G2 = (V2;E2). The union of G1 and G2, denoted by G1 [ G2, is the graph that has as set of vertices V1 [ V2 and as set of edges E1 [ E2. The product of G1 and G2, denoted by G1 × G2, is the graph that has as set of vertices V1 × V2 and whose adjacencies are given by (u1; u2) ∼ (v1; v2) , (u1v1 2 E1 and u2 = v2) or (u1 = v1 and u2v2 2 E2): 1.2. Exercises 3 1.2 Exercises 1.1 For each of the graphs Nn, Kn, Pn, Cn and Wn, give: 1) a drawing for n = 4 and n = 6; 2) the adjacency matrix for n = 5; 3) the order, the size, the maximum degree and the minimum degree in terms of n. 1.2 For each of the following statements, find a graph with the required property, and give its adjacency list and a drawing. 1) A 3-regular graph of order at least 5. 2) A bipartite graph of order 6. 3) A complete bipartite graph of order 7. 4) A star graph of order 7. 1.3 Find out whether the complete graph, the path and the cycle of order n ≥ 1 are bipartite and/or regular. 1.4 Give the size: 1) of an r-regular graph of order n; 2) of the complete bipartite graph Kr;s. 1.5 Let V = fa; b; c; d; e; fg, E = fab; af; ad; be; de; efg and G = (V; E). Determine all the subgraphs of G of order 4 and size 4. 1.6 The following five items refer to the graph G defined as follows. The set of vertices is V = f0; 1; 2; 3; 4; 5; 6; 7; 8g, and two vertices u and v are adjacent if ju − vj 2 f1; 4; 5; 8g. Determine the order and the size of the following subgraphs of G: 1) The subgraph induced by even vertices. 2) The subgraph induced by odd vertices. 3) The subgraph induced by the set f0; 1; 2; 3; 4g. 4) A spanning subgraph with as many edges as possible but without cycles. 1.7 Consider the graph G = (V; E) with V = f1; 2; 3; 4; 5g and E = f12; 13; 23; 24; 34; 45g. Give the set of edges, the incidence and adjacency matrices, and a drawing of the graphs Gc, G − 4, G − 45 and G + 25. 4 Chapter 1. Graphs: basic concepts 1.8 Consider a graph G = (V; E) of order n and size m. Let v be a vertex and e an edge of G. Give the order and the size of Gc, G − v and G − e. 1.9 Find out whether the complement of a regular graph is regular, and whether the comple- ment of a bipartite graph is bipartite. If so, prove it; if not, give a counterexample. 1.10 Give the set of edges and a drawing of the graphs K3 [ P3 and K3 × P3, assuming that the sets of vertices of K3 and P3 are disjoint. 1.11 Consider the graphs G1 = (V1;E1) and G2 = (V2;E2). Give the order, the degree of the vertices and the size of G1 × G2 in terms of those of G1 and G2. 1.12 Prove or disprove the following statements: 1) If G1 and G2 are regular graphs, then G1 × G2 is regular. 2) If G1 and G2 are bipartite graphs, then G1 × G2 is bipartite. 1.13 Draw all the graphs that have V = fa; b; cg as set of vertices. 1.14 Consider graphs whose set of vertices is [7] = f1; 2; 3; 4; 5; 6; 7g. Compute how many of them are there . 1) . with exactly 20 edges. 2) . with exactly 16 edges. 3) . in total. 1.15 For each of the following sequences, find out if there is any graph of order 5 such that the degrees of its vertices are given by that sequence. If so, give an example. 1)3 ; 3; 2; 2; 2. 3)4 ; 3; 3; 2; 2. 5)3 ; 3; 3; 3; 2. 2)4 ; 4; 3; 2; 1. 4)3 ; 3; 3; 2; 2. 6)5 ; 3; 2; 2; 2. 1.16 Prove that if a graph is regular of odd degree, then it has even order. 1.17 Let G be a bipartite graph of order n and regular of degree d ≥ 1. Which is the size of G? Could it be that the order of G is odd? 1.18 Prove that the size of a bipartite graph of order n is at most n2=4. 1.19 Let G be a graph with order 9 so that the degree of each vertex is either 5 or 6. Prove that there are either at least 5 vertices of degree 6 or at least 6 vertices of degree 5. 1.2. Exercises 5 1.20 Alex and Leo are a couple, and they organize a party together with 4 other couples. There are a number of greetings but, naturally, nobody says hello to their own partner.