Combinatorics Summary Sheet for Exam 1 Material 2019

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1 Combinatorics Summary Sheet for Exam 1 Material Graphs Graph An ordered three-tuple (V, E, F ) where V is a set representing the vertices, E is a set representing the edges, and F is a function F : E ( V 2) V. More simply, F is a function mapping edges to unordered pairs or single vertices (for self-loops). Simple Graph A graph such that Image(F ) ( V 2) and F is an injection. More simply, there are no self-loops, and each pair of vertices can have at most one edge between them. Parallel Edges e 1 and e 2 are parallel edges if they are non-loops with the same end points. Directed Graph/Digraph A graph such that F : E V V. More simply, F takes edges and returns ordered pairs. Simple Digraph There are no loops, and at most one edge in each direction between v 1 and v 2 if v 1 v 2. Isomorphic Consider two simple graphs G 1 and G 2. We say that G 1 and G 2 are isomorphic if and only if: There exists a bijection F : V (G 1 ) V (G 2 ) such that V 1 V 2 E(G 1 ) if and only if F (V 1 )F (V 2 ) E(G 2 ) Adjacency We say that two vertices v 1 v 2 are adjacent iff there exists an edge with endpoints v 1 and v 2. Kn Any graph with n vertices and exactly one edge between every pair of vertices. Km, n Here is an example of K 3,3 (labeled K 3 in the picture): Pn These are sometimes called the polygon graphs: Γ(v) The set of all vertices to which v is adjacent, excluding itself. More formally, Γ(v) = {w V : w v and w is adjacent to v} deg(v) The number of edges incident to v. Note that loops add 2 to the degree of a vertex. Regular A graph where all vertices have the same degree. First Theorem of Graph Theory For any finite graph G, deg(v) = 2 E(G). v V (G) 1

2 This means the sum of the degrees of every vertex of G is equal to twice the number of edges in G. Theorem 1.1 Every finite graph has an even number of vertices of odd degree. Walk A sequence of vertices interspersed with edges, of the form where the endpoints of e i are exactly v i 1 and v i. Path A walk with distinct edges. v 0, e 1, v 1,..., e k, v k Simple Path A walk with distinct vertices. This property forces distinct edges as well. Simple Closed Path A simple path in which v 0 = v k, and k 1. Subgraph A graph H is a subgraph of G iff V (H) V (G), E(H) E(G), and the ends of an edge e E(H) are the same as its ends in G. Spanning Subgraph A subgraph in which all vertices in G are also in H. A spanning subgraph does not need to use all edges in G. Induced Subgraph A subgraph in which all edges in G that are between vertices in H are in H as well. Is connected by a path to An equivalence relation on V (G), where two vertices are related if there is a path that connects them. Component The induced subgraphs on the equivalence classes of the is connected by a path to relation. Connected A graph in which there is a path between every pair of vertices. In other words, a graph with exactly one component. Maximally Connected A graph H is a maximally connected subgraph of G if it is 1. A subgraph of G. 2. Connected. 3. Not a proper subgraph of a connected subgraph of G. In other words, we can t add any more vertices or edges to H and maintain a connected subgraph. Note that the textbook defines a component as a maximally connected subgraph. d(v i, v j ) is the length of the shortest path between v i and v j. If no path between v i and v j exists, then d(v i, v j ) =. Tree A connected graph with no polygons (P n ) as subgraphs. Eulerian A graph G is Eulerian if there exists a closed walk using every edge of G exactly once. By this definition, graphs with isolated vertices can be Eulerian. Theorem 1.2 Let G be a finite graph without isolates (vertices of degree 0). G is Eulerian if and only if G is connected and every vertex of G has even degree. 2

3 2 Trees Hamiltonian Circuit A simple closed path that passes through each vertex in the graph exactly once. A graph is Hamiltonian iff it has a polygon as a spanning subgraph. [n] The set of all positive integers up to and including n. [n] n 2 The set of sequences of length n 2, where the numbers in each sequence are chosen from [n]. Theorem 2.1 There are n n 2 different labeled trees on n vertices. 2.1 Labeled Tree to Prüfer Code The function δ : Labeled trees on [n] [n] n 2 takes a labeled tree and returns its unique Prüfer code. Given a tree on n vertices labeled with [n], we can find its Prüfer code δ(t ) with the following algorithm: 1. Find the lowest labeled leaf. 2. Write down the label of the vertex to which the leaf is adjacent. 3. Delete the leaf and its incident edge. 4. Repeat steps 1-3 until only 2 vertices remain. 2.2 Prüfer Code to Labeled Tree Note that the leaves of T do not appear in δ(t ). Additionally, the number of times that a label v appears in the code is deg(v) 1. Because of these qualities, δ is actually a bijection, and has an inverse δ 1 : [n] n 2 Labeled trees on [n]. So given a Prüfer code, we can find the exact tree that corresponds to it. Given a Prüfer code s from [n] n 2, we can find the corresponding tree δ 1 (s) with the following algorithm: 1. Write down s. Next to it, write down the set {1, 2,..., n}. 2. Write down the smallest number in the set not in the sequence. 3. Delete that number from the set and delete the first number in the sequence. 4. Repeat steps 2 3 until there are no numbers left in the sequence. Minimum Weight Spanning Trees: Consider a finite connected graph G with real numbers assigned to each edge. We want to find a spanning subtree of G such that the sum of the edge labels is as small as possible. Such a subtree is called a minimum weight spanning tree, or MWST. Given a finite connected graph G with n vertices, we can find a MWST with the following algorithm: 1. Choose an edge for the MWST by selecting the lightest edge in G that has not yet been selected and does not create a cycle (polygon) with previously selected edges. 3

4 2. Repeat step 1 until you have a tree with edges e 1, e 2,..., e n 1 where edges are labeled by the order in which they were selected. Note that MWSTs are not unique, and that any MWST for G can be the result of this algorithm. 3 Colorings of Graphs and Ramsey s theorem Let G be a finite simple graph. A proper coloring of G is a function F : V (G) [n] such that F (u) F (v) whenever u and v are endpoints of the same edge. The chromatic number, χ(g), is the smallest n for which a proper coloring of G exists. A graph G is bipartite iff χ(g) = 1 or χ(g) = 2. Bipartiteness Theorem A graph G is bipartite if and only if G has no odd cycles (polygons). Four Color Theorem If G is a planar graph, then χ(g) 4. This holds for infinite graphs, as well as finite graphs, assuming the axiom of choice (Theorem of Erdős and De Bruijn). A Greedy Algorithm: Consider a graph G such that deg(v) d for all v V (G). Then χ(g) d + 1. To find a coloring of G with d + 1 or fewer colors, we can use the following greedy algorithm: 1. Order the vertices of G in any order. 2. Go through the vertices in order and assign to each vertex the smallest value in [d + 1] that has not already been assigned to one of its neighbors. Continue until all vertices are colored. Theorem 3.1 (Brooks Theorem) Fix d 3. Let G be a graph in which deg(v) d for all v V (G). If G does not have K d+1 as a subgraph, then χ(g) d. A Formula for Monochromatic Triangles: Color the edges of K n red or blue. Let T i be the number of induced 3-vertex subgraphs of K n having exactly i red edges. Note that ( ) n T 0 + T 1 + T 2 + T 3 = 3 This is true, as T 0, T 1, T 2 and T 3 represent all possible induced 3-vertex subgraphs. We can also see that 3T 0 + T 1 + T 2 + 3T 3 = [ ( ) ( ) d r (v) db (v) ] v V (K n) v V (K n) where d r (v) is the number of red edges incident with v and d b (v) is the number of blue edges incident with v. Together, these two facts imply that T 3 + T 0 = 1 (d 2 4 r(v) + d 2 b(v)) 1 ( ) n 1 ( ) n

5 Theorem 3.3 Let r 1 and q 1, q 2 r be given. There exists a smallest positive integer N(q 1, q 2 ; r) such that for every n N(q 1, q 2 ; r) and each 2-coloring of the r-element subsets of [n], there exists either a q 1 -element subset such that all of its r-element subsets are colored 1 or a q 2 -element subset such that all of its r-element subsets are colored 2. Theorem 3.4 N(p, q; 2) ( ) p+q 2 p 1. Theorem 3.5 N(p, p; 2) 2 p/2. Infinite Ramsey Theorem Consider any coloring of [N] r with a finite number of colors. There exists some Y N such that [Y ] r is monochromatic and Y =. Theorem 3.8 (Happy Ending Problem) For any n, there exists an integer N(n) such that in any collection of N N(n) points in the plane, no 3 collinear, there is subset of n points forming a convex n-gon. 4 Turán s Theorem Theorem 4.1 (Turan) more than edges, then G has K p as a subgraph. Let n, p N, with n = t(p 1) + r and 1 r p 1. If G is a simple graph with M(n, p) = p 2 r(p 1 r) 2(p 1) n2 2(p 1) Theorem 4.2 If G is a simple graph on n vertices with no P 3 or P 4, then G has at most 1 2 n n 1 edges. Theorem 4.3 a P n. If G is a simple graph with n 3 vertices such that deg(v) n 2 for all v V (G), then G has 5 System s of Distinct Representatives Matching A matching in a graph G is a set of edges such that no pair of edges share an endpoint. Complete Matching Let G be a bipartite graph with bipartition X Y, where X, Y. A complete matching from X to Y is a matching of G such that every vertex in X is matched. Γ(S) Given a subset S X, we define Γ(S) as Γ(S) = x S Γ(x) 5

6 Hall s Condition We say that Hall s Condition holds for X if and only if for all subsets S X, Γ(S) S. Theorem 5.1 (Hall) Let G be a bipartite graph with bipartition X Y, where X, Y. There exists a complete matching from X to Y if and only if Hall s Condition holds for X. Theorem 5.4 (König) The minimum number of lines needed to cover the 1 s of a (0,1)-matrix A is equal to the size of a maximum independent set of 1 s in A. Theorem 5.5 (Birkhoff) Any n n matrix of non-negative integers with all rows and columns summing to L > 0 can be written as a sum of L permutation matrices. 6 Dilworth s Theorem and extremal set theory order relation is an order relation on P is reflexive, transitive, and antisymmetric. We usually write instead of. If we are considering multiple order relations on a set, then we can distinguish them by writing 1, 2, etc. chain C is a chain x, y C we have x y or y x. The elements are pairwise comparable. antichain A is an antichain x, y C we have x y and y x. The elements are pairwise incomparable. Theorem 6.1 (Dilworth) If P is finite, then the minimum number of chains necessary to partition P is equal to the the size of a maximum antichain contained in P. Theorem 6.2 (Mirsky) If P has no chain of size M + 1, then P is a union of M antichains. Theorem 6.4 Let A = {A 1,..., A m } be a set of k-element subsets of [n], (2k n), and suppose that A i A j for all i, j [m]. Then m ( n 1 k 1). 6