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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 v 1 ,

.. ., vn, edges v 1 v 2 , v 2 v 3 ,.. ., vn− 1 vn, vnv 1 ).

Definition 1.6. Path on n-vertices, Pn; defined by picture (vertices v 1 ,.. ., vn, edges v 1 v 2 , v 2 v 3 ,.. ., vn− 1 vn).

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.

1.2 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 u, v∈V (G)

d(u, v).

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) ≤ 2 m.

Definition 1.17. Let G be a graph with vertices v 1 ,.. ., vn. The adjacency matrix of G is the n × n matrix whose (i, j) entry, denoted by [A]i,j , is defined by

[A]i,j =

1 , if vi and vj 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 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 A^2 = A · A and Ak^ = Ak−^1 · A for k ≥ 2.

Theorem 1.6. Let G be a graph with vertices v 1 , v 2 ,.. ., vn and correspond- ing adjacency matrix A. Then for any positive integer k, the (i, j) entry of Ak^ equals the number of walks from vi to vj that use exactly k edges.

Corollary: Let G be a graph with vertices v 1 , v 2 ,.. ., vn and corresponding adjacency matrix A. If d(vi, vj ) = m, then [Ak]i,j = 0 for 1 ≤ k < m.

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

Sk = I + A + A^2 + · · · + Ak

where I is the identity matrix:

I =

.

Theorem 1.7. Let G be a graph with vertices v 1 ,.. ., 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 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 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 a 1 , a 2 ,

.. ., 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

[D]i,j =

deg(vi), if 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+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.

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 K 1 , 3 (claw), Z 1 and N. See book for pictures.

Theorem 1.18. A 2 -connected, {K 1 , 3 , Z 1 }-free graph is Hamiltonian.

Theorem 1.19. 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 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 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 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.

Theorem 1.22. The graph K 3 , 3 is not planar.

Theorem 1.23. Let G be a planar graph with n ≥ 3 vertices and q edges. Then one has q ≤ 3 n − 6 and if equality holds, every region is bounded by three edges.

Theorem 1.24. The graph K 5 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 K 3 , 3 or K 5.

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).

Chapter 2

Combinatorics