Discrete Mathematics (Math 510) — Fall 2014

Definitions and Theorems

Gerald Hoehn

October 12, 2014 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 Kn. Definition 1.3. A empty graph is a graph without edges. The empty graph of order n is denoted by En. 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, Cn; defined by picture (vertices v1, ..., vn, edges v1v2, v2v3, ..., vn−1vn, vnv1).

Definition 1.6. Path on n-vertices, Pn; defined by picture (vertices v1, ..., vn, edges v1v2, v2v3, ..., vn−1vn). 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 S 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 “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 1.10. 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 1.11. A walk in a graph is a sequence of vertices v1, v2, ..., vk such that vivi+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 v1, ..., vk where k ≥ 3 and vkv1 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 1.12. 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 1.13. 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 1.14. The connectivity κ(G) of a graph G is the minimal num- ber of vertices v1, ..., vk one has to choose such that G − {v1,...,vk} 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 1.2 Distance

General assumption in this section: G is connected.

Definition 1.15. 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 dG(u,v) or just d(u,v).

This implies that d(u,u) = 0 for any vertex u.

Definition 1.16. The diameter of a graph G, denoted diam(G), is the greatest distance between any two vertices:

diam(G)= max d(u,v). u,v∈V (G)

Theorem 1.4. On has:

(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 1.17. Let G be a graph with vertices v1, ..., vn. 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 vj are adjacent, [A]i,j = 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 1.18. 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 A2 = A A and Ak = Ak−1 A for k ≥ 2.

Theorem 1.6. Let G be a graph with vertices v1, v2, ..., vn and correspond- ing adjacency matrix A. Then for any positive integer k, the (i,j) entry of k A equals the number of walks from vi to vj that use exactly k edges.

Corollary: Let G be a graph with vertices v1, v2, ..., vn and corresponding k adjacency matrix A. If d(vi,vj)= m, then [A ]i,j = 0 for 1 ≤ k < m.

3 Definition 1.19. For an n × n matrix A and a positive integer k set

2 k Sk = I + A + A + + A where I is the identity matrix: 1 0 ... 0 0  0 1 ... 0 0  . . I =  . .     0 0 ... 1 0     0 0 ... 0 1   . Theorem 1.7. Let G be a graph with vertices v1, ..., vn. If m is the smallest positive integer such that all entries of Sm are positive, then diam(G)= m.

1.3 Trees

Definition 1.20. . • A forest is a graph containing no cycles as subgraphs. • A 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 1.10. 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 nn−2 distinct labeled trees of order n.

Definition 1.21. A Pruefer seqence of a labeled tree is a sequence a1, a2, ..., an−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 1.22. 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 1.23. Given an n × n-matrix M, the (i,j)-cofactor of M is defined to be (−1)i+jdet(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 Theorem 1.12 (Matrix Tree Theorem (Kirchhoff)). Let G be a connected labeled graph with adjacency matrix A and degree matrix D. Then the num- ber 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 1.24. 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 1.25. 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 1.13. For a connected graph G the following statements are equiv- alent:

(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 1.14. 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 1.26. Let G be a graph. A (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 1.16. 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 1.27. 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 1.17. Let G be a connected graph of order ≥ 3 with vertex con- nectivity κ(G) and independence number α(G). If κ(G) ≥ α(G), then G is Hamiltonian.

5 Definition 1.28. 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 1.29. 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 K1,3 (claw), Z1 and N. See book for pictures.

Theorem 1.18. A 2-connected, {K1,3, Z1}-free graph is Hamiltonian.

Theorem 1.19. Let G be a {K1,3, N}-free graph. 1. If G is connected, then G is traceable.

2. If G is 2-connected, then G is Hamiltonian.

Definition 1.30. The line graph L(G) of a graph G is defined in the fol- lowing 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 corre- sponding edges in G share a vertex.

Problem 1.4.3. # 11: If G is Eulerian, then L(G) is Hamiltonian.

1.5 Planarity

Definition 1.31. A 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 1.32. 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 1.33. 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 1.20. A forest is a planar graph.

Theorem 1.21 (Euler’s Formula). For a planar representation of a con- nected 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 Theorem 1.22. The graph K3,3 is not planar. Theorem 1.23. 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 1.24. The graph K5 is not planar. Theorem 1.25. If G is a planar graph, then G contains a vertex of degree at most five, i.e. δ(G) ≤ 5.

Regular Polyhedrons Definition 1.34. 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 1.26. 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 1.35. A polyhedral graph is a planar graph satisfying properties (i), (ii) and (iii) from Proposition 1.26. Theorem 1.27. Let G be a polyhedral graph. Then there exists a region with boundary degree at most 5. Definition 1.36. A regular convex polyhedron is a convex polyhedron with congruent regular polygons as faces which are assembled the same way around each vertex. Theorem 1.28. Let G be the planar graph associated to a regular con- vex 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 1.29. 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 1.37. 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 1.30. 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 K3,3 or K5.

7 1.6 Colorings

Definition 1.38. Let G be a graph and k be a positive integer. A k-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 1.39. 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 1.32. Let G be a graph. Then χ(G) ≤ ∆(G) + 1, where ∆(G) is the maximal degree of G.

Definition 1.40. 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 1.33. For a graph G one has χ(G) ≥ ω(G).

Theorem 1.34. Let G be a planar graph. Then G is 5-colorable.

Theorem 1.35 (Apel & Haken (1976)). Every planar graph is 4-colorable.

Theorem 1.36. Let G be a graph with no cycles of odd order. Then G is 2-colorable.

Theorem 1.37. 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 1.41. 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 Definition 1.42. 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 1.38. A matching M in a graph G is a maximum matching if and only if there exist no M-augmenting paths in G.

Definition 1.43. 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 1.44. 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 1.45. Let G be a graph. An edge cover C of G is a set of vertices such that every edge of G is incident with at least one vertex from C.

Theorem 1.40 (K¨onig-Egerv`ary 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 1.46. 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 Chapter 2

Combinatorics

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