Discrete Mathematics (Math 510) Fall Definitions and Theorems

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1 Discrete Mathematics (Math 510) Fall 2014 Definitions and Theorems Gerald Hoehn October 12, 2014

2 Chapter 1 Graph Theory 1.1 Basics Definition 1.1. A graph G is an ordered pair (V,E) of disjoint finite sets such that E is a subset of the set of unordered pairs of V. We call V = V(G) the vertices of G and E = E(G) the edges of G. An edge {x, y} in E is said to join the vertices x and y and denoted by xy. We call x and y the end-vertices of the edge xy. Two vertices x and y are called adjacent if xy is an edge of G. The order of G is the cardinality of V, the size of G is the cardinality of E. Definition 1.2. A complete graph is a graph in which all distinct vertices are adjacent. The complete graph of order n is denoted by K n. Definition 1.3. A empty graph is a graph without edges. The empty graph of order n is denoted by E n. Definition 1.4. The complement G of a graph G is the graph with the same vertex set as G, and whose edge set consists of all edges not present in G. Definition 1.5. Cycle on n vertices, C n ; defined by picture (vertices v 1,..., v n, edges v 1 v 2, v 2 v 3,..., v n 1 v n, v n v 1 ). Definition 1.6. Path on n-vertices, P n ; defined by picture (vertices v 1,..., v n, edges v 1 v 2, v 2 v 3,..., v n 1 v n ). Definition 1.7. A graph G is bipartite if its vertex set can be partitioned into two sets X and Y such that every edge of G has one end vertex in X and the other in Y. The complete bipartite graph K X, Y is the graph with vertex set V = X Y and edge set E = {xy x X, y Y}. Definition 1.8. Two graphs G and H are said to be isomorphic if there exists a one-to-one correspondence f : V(G) V(H) such for each pair of vertices x, y V(G) one has xy E(G) if and only if f(x)f(y) E(H). One calls f an isomorphism. 1

3 Isomorphic is a precise way to say that we ignore the labels of the vertices. Definition 1.9. We say that a graph H is a subgraph of a graph G if V(H) V(G) and E(H) E(G). We also say that H is a subgraph if H is isomorphic to a subgraph in the above sense. Definition For G a graph and v a vertex the degree deg(v) is the number of edges with end-vertex v. We let δ(g) be the minimal and (G) be the maximal degree of any vertex of G. Theorem 1.1. The sum of the degrees of the vertices is equal to twice the numbers of edges: deg(v) = 2 E(G). v V(G) Definition A walk in a graph is a sequence of vertices v 1, v 2,..., v k such that v i v i+1 is an edge for each i = 1,..., k 1. A path is a walk in which all vertices are distinct. A cycle is path v 1,..., v k where k 3 and v k v 1 is also an edge. The length of a walk, path or cycle is the number of corresponding edges. Theorem 1.2. If in a graph there exists a walk from u to v, then there exists also a path from u to v. Definition Let G be a graph. For an edge e of G, G e is the graph with the same vertices and all edges besides e. For an vertex v of G, G v is the graph obtained by removing v and all edges ending in v. Definition A graph is called connected if every two vertices can be joined by a path. Otherwise it is called disconnected. Note that every graph is the disjoint union of connected components. Definition The connectivity κ(g) of a graph G is the minimal number of vertices v 1,..., v k one has to choose such that G {v 1,..., v k } is disconnected. A graph is called k-connected if k κ(g). Note that a graph is connected if and only if κ(g) 1, i.e., G is 1- connected. Theorem 1.3. A graph is bipartite if and only if it contains no cycle of odd length. 2

4 1.2 Distance General assumption in this section: G is connected. Definition Let u, v be two vertices of a graph G. The distance between u and v is the length of the shortest path connecting u with v and denoted by d G (u,v) or just d(u,v). This implies that d(u,u) = 0 for any vertex u. Definition The diameter of a graph G, denoted diam(g), is the greatest distance between any two vertices: Theorem 1.4. On has: diam(g) = max u,v V(G) d(u,v). (i) d(u,v) 0 for all u, v and d(u,v) = 0 if and only if u = v. (ii) d(u,v) = d(v,u) for all u, v. (iii) d(u,w) d(u,v)+d(v,w) for all u, v, w. Theorem 1.5. Let u be any vertex in a graph G. Let m be the maximal distance between u and any other vertex. Then m diam(g) 2m. Definition Let G be a graph with vertices v 1,..., v n. The adjacency matrix of G is the n n matrix whose (i,j) entry, denoted by [A] i,j, is defined by { 1, if vi and v [A] i,j = j are adjacent, 0, otherwise. For graphs (without loops) the entries on the diagonal of A are zero. The matrix is symmetric, i.e., [A] i,j = [A] j,i for all i and j. Definition Let A and B two n n matrices. The product matrix A B is the n n matrix with entries [A B] i,j = [A] i,1 [B] 1,j +[A] i,2 [B] 2,j + +[A] i,n [B] n,j. We set shortly A 2 = A A and A k = A k 1 A for k 2. Theorem 1.6. Let G be a graph with vertices v 1, v 2,..., v n and corresponding adjacency matrix A. Then for any positive integer k, the (i,j) entry of A k equals the number of walks from v i to v j that use exactly k edges. Corollary: Let G be a graph with vertices v 1, v 2,..., v n and corresponding adjacency matrix A. If d(v i,v j ) = m, then [A k ] i,j = 0 for 1 k < m. 3

5 Definition For an n n matrix A and a positive integer k set where I is the identity matrix: S k = I +A+A 2 + +A k I = Theorem 1.7. Let G be a graph with vertices v 1,..., v n. If m is the smallest positive integer such that all entries of S m are positive, then diam(g) = m. 1.3 Trees Definition A forest is a graph containing no cycles as subgraphs. A tree is a connected forest. A leave is a vertex of degree 1 in a tree. Theorem 1.8. A tree T of order n has n 1 edges. Theorem 1.9. A graph of order n is a tree if and only if it is connected and contains n 1 edges. Theorem A graph of order n is a tree if and only if it contains no cycles and contains n 1 edges. Theorem 1.11 (Cayley s Tree formula). There are n n 2 distinct labeled trees of order n. Definition A Pruefer seqence of a labeled tree is a sequence a 1, a 2,..., a n 2 of numbers from {1, 2,..., n} assigned in a unique way to a tree. (See the book for the precise algorithm.) A labeled tree can be uniquely reconstructed from its Pruefer sequence. (See the book for the precise algorithm.) Definition The degree matrix of a graph G is the n n matrix D defined by { deg(vi ), if i = j, [D] i,j = 0, else. Definition Given an n n-matrix M, the (i,j)-cofactor of M is defined to be ( 1) i+j det(m(i j)) where det(m(i j)) is the determinant of the (n 1) (n 1)-matrix one gets by deleting the i-th row and the j-th column from M. 4.

6 Theorem 1.12 (Matrix Tree Theorem (Kirchhoff)). Let G be a connected labeled graph with adjacency matrix A and degree matrix D. Then the number of unique spanning trees of G is equal to the value of any cofactor of the matrix D A. 1.4 Eulerian and Hamiltonian walks Eulerian trails and circuits Definition Let G be a graph. A trail in G is a walk that does not repeat any edges. A closed trail is called a circuit. A cycle is a circuit. Definition An Eulerian trail (resp. Eulerian circuit) in a graph G is a trail (resp. circuit) in G which includes every edge. A graph that contains an Eulerian circuit is said to be an Eulerian graph. Theorem For a connected graph G the following statements are equivalent: (1) G is Eulerian. (2) Every vertex of G has even degree. (3) The edges of G can be partitioned into (edge disjoint) cycles. Corollary A connected graph contains an Eulerian trail if and only if there are at most two vertices of odd degree. Hamiltonian paths and cycles Definition Let G be a graph. A Hamiltonian path (resp. Hamiltonian cycle) in G is a path (resp. cycle) which visits each vertex exactly once. A graph that contains a Hamiltonian cycle (resp. Hamiltonian path) is said to be a Hamiltonian graph (resp. traceable graph). Theorem 1.15 (Dirac). Let G be a graph of order n and minimal degree δ(g) n/2. Then G is Hamiltonian. Theorem Let G be a graph of order n. If for any pair of nonadjacent vertices x and y one has deg(x)+deg(y) n, then G is Hamiltonian. Definition A set of vertices in a graph is called independent if they are pairwise nonadjacent. The independence number α(g) of a graph G is the maximal number of independent vertices in G. Theorem Let G be a connected graph of order 3 with vertex connectivity κ(g) and independence number α(g). If κ(g) α(g), then G is Hamiltonian. 5

7 Definition Let G be a graph and W be a subset of the set of vertices of G. The graph with vertex set W and edges as in G is called the subgraph induced by W. Definition Let G and H be two graphs. The graph G is said to be H-free if G does not contain a copy of H as induced subgraph. Examples of graphs K 1,3 (claw), Z 1 and N. See book for pictures. Theorem A 2-connected, {K 1,3, Z 1 }-free graph is Hamiltonian. Theorem Let G be a {K 1,3, N}-free graph. 1. If G is connected, then G is traceable. 2. If G is 2-connected, then G is Hamiltonian. Definition The line graph L(G) of a graph G is defined in the following way: The vertices of L(G) are the edges of G. Two vertices of L(G) are joined by an edge if and only if the corresponding edges in G share a vertex. Problem # 11: If G is Eulerian, then L(G) is Hamiltonian. 1.5 Planarity Definition A planar graph is a graph that can be drawn in the plane in such a way that pairs of edges intersect only at vertices, if at all. A drawing of a planar graph G in the plane as above is called a planar representation. Definition Given a planar representation of a graph G, a region is a maximal set of the plane in which any two points can be joined by a curve that does not intersect in any part of G. The exterior region is the region containing arbitrary far away points. Definition Given a region R in a planar representation of a graph, the boundary degree of R, denoted by b(r), is the number of edges that bound the region R. Theorem A forest is a planar graph. Theorem 1.21 (Euler s Formula). For a planar representation of a connected graph G with n vertices, q edges, and r regions one has n q +r = 2. In particular, the number of regions is independent of the chosen planar representation of a planar graph. 6

8 Theorem The graph K 3,3 is not planar. Theorem Let G be a planar graph with n 3 vertices and q edges. Then one has q 3n 6 and if equality holds, every region is bounded by three edges. Theorem The graph K 5 is not planar. Theorem If G is a planar graph, then G contains a vertex of degree at most five, i.e. δ(g) 5. Regular Polyhedrons Definition A convex polyhedron is a three dimensional solid which is the intersection of finitely many half-spaces such that the boundary consists of faces, edges and corners (also called vertices). Proposition By declaring one face as the exterior region, the vertices and edges of a convex polyhedron define a planar representation of a graph with the following properties: (i) Every vertex has degree greater or equal to 3. (ii) Every edge is bounded by exactly 2 regions. (iii) Every region is bounded by at least 3 edges. Definition A polyhedral graph is a planar graph satisfying properties (i), (ii) and (iii) from Proposition Theorem Let G be a polyhedral graph. Then there exists a region with boundary degree at most 5. Definition A regular convex polyhedron is a convex polyhedron with congruent regular polygons as faces which are assembled the same way around each vertex. Theorem Let G be the planar graph associated to a regular convex polyhedron. Then the numbers (n, q, r) of vertices, edges and regions of G belong to the five values (4,6,4), (8,12,6), (6,12,8), (12,30,20) and (20, 30, 12). Theorem For each of the 5 cases (n,q,r) of Theorem 1.28 there exists a unique regular convex polyhedron with associated graph G. Definition Let G be a graph. A subdivision of an edge e in G is a substitution of a path for e. A graph H is called a subdivision of G if H can be obtained from G by a finite number of subdivisions of edges of G. Theorem A graph G is planar if and only if an arbitrary subdivision of G is planar. Theorem 1.31 (Kuratowski (1930)). A graph G is planar if and only if it contains no subdivision of K 3,3 or K 5. 7

9 1.6 Colorings Definition LetGbe agraph and k be apositive integer. Ak-coloring of G is a function K : V(G) C from the vertex set into a set C of size k called colors. A proper k-coloring of G is a k coloring K such that K(u) K(v) for all adjacent vertices u and v. If such a coloring exists for a graph G, we say that G is k-colorable. Usually one takes C = {1, 2,..., k}. Definition The chromatic number of a graph G is the smallest integer k such that G is k-colorable. It is denoted by χ(g). For any graph G of order n one has χ(g) n. Theorem Let G be a graph. Then χ(g) (G)+1, where (G) is the maximal degree of G. Definition The clique number of a graph G, denoted by ω(g), is defined as the order of the largest complete graph contained as a subgraph in G. Theorem For a graph G one has χ(g) ω(g). Theorem Let G be a planar graph. Then G is 5-colorable. Theorem 1.35 (Apel & Haken (1976)). Every planar graph is 4-colorable. Theorem Let G be a graph with no cycles of odd order. Then G is 2-colorable. Theorem Let G be a planar graph with a planar representation in which each region has even boundary degree. Then G is 2-colorable. 1.7 Matchings Definition Let G be a graph. A matching M in G is a set of edges of G such that no two edges in M are adjacent to a common vertex. A maximal matching is a matching in G which cannot be extended to a larger matching by additional edges. A maximum matching is a matching in G whose number of edges is maximal among all matchings in G. A perfect matching is a matching in G such that all vertices in G are incident to an edge of the matching. 8

10 Definition Let M be a matching in a graph G. An M-alternating path is a path in G such that the edges alternate between edges in M and edges not contained in M (or vice versa). An M-augmenting path is an M-alternating path such that its two ends a not incident with any edge from M. Theorem A matching M in a graph G is a maximum matching if and only if there exist no M-augmenting paths in G. Definition Let G be a bipartite graph with respect to the vertex sets X and Y. One says X can be matched into Y if there exists a matching in G that contains all vertices of X. Definition Let G be a graph. For a set S V(G), we let N(S) be the set of all vertices of G which are connected to a vertex S by an edge. Theorem 1.39 (Marriage Theorem (Hall)). Let G be a bipartite graph with partition sets X and Y. Then X can be matched into Y if and only if N(S) S for all subset S of X. Definition LetGbeagraph. Anedge cover C ofgisasetofvertices such that every edge of G is incident with at least one vertex from C. Theorem 1.40 (König-Egervàry Theorem). Let G be a bipartite graph. Then the maximum number of edges in a matching in G equals the minimum number of vertices in an edge cover of G. Theorem 1.41 (Max flow min cut theorem). The maximum value of the total flow of any water flow in a graph equals the minimum capacity of a cut. Definition For a graph G, denote by Ω(G) the number of connected components of odd order. Theorem 1.42 (Generalized Marriage Theorem (Tutte 1947)). Let G be a graph of order 2. Then G has a perfect matching if and only if Ω(G S) S for all subsets S of the vertex set V(G). 9

11 Chapter 2 Combinatorics 10

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