Undergraduate Texts in Mathematics

At the beginning of this chapter, a new section focuses on multinomial coefficients and their properties, following the development of the binomial coefficients. Begin at the beginning,” said the king gravely, “and go on until you come to the end; then stop."

Graph Theory

Introductory Concepts

Graphs and Their Relatives

The field of graph theory began to flourish in the twentieth century as more and more modeling possibilities were recognized – and the growth continues. Adam's list: Doug Bert's list: Adam, Ernie Chuck's list: Doug, Ernie Doug's list: Chuck Ernie's list: Ernie Frank's list: Adam, Bert.

FIGURE 1.3. A digraph.

The Basics

In simple graphs, this is the same as the (open) neighborhood cardinality ofv. If is a rank graph, what is the maximum number of edges in G. Prove that for any rank graphGof at least 2, the degree sequence has at least one pair of repeated entries.

FIGURE 1.8. Deletion operations.

Special Types of Graphs

Again, the graphs in Figure 1.22 show that the opposite of this statement is not true. Prove that if the graphs GandH are isomorphic, then their complements G. Prove that the two graphs in Figure 1.24 are not isomorphic.

FIGURE 1.11. Examples of complete graphs.

Distance in Graphs

Definitions and a Few Properties

Perhaps you'd prefer your favorite graph to avoid the limelight and stay on the fringes. Not only does this mean that G is complete, it also means that every vertex of G is on the periphery. G is the periphery of itself.

Graphs and Matrices

The matrixM is called the distance matrix of graphG. Give the adjacency matrix for each of the following graphs. a) P2k and P2k+1, where the vertices from one end of the path to the other are labeled. Let be the adjacency matrix for the graph G. a) Show that the number of triangles containing vji is 12[A3]j,j. b) The trace of a square matrixM, denoted Tr(M), is the sum of the.

Graph Models and Distance

A researcher's Erd˝os number is thus the distance from the corresponding peak to the Erd˝os peak. Under these more open guidelines, mathematician Paul Erd˝os actually has a Bacon number of 3.

Trees

Definitions and Examples

The leaves on the tree in Figure 1.32 correspond to the outcomes in the probability space for this experiment. The bonds between carbon and hydrogen atoms are depicted in the trees in Figure 1.34.

FIGURE 1.31. Trees of order 6 or less.

Properties of Trees

In fact, the stump (K1) is the only such tree, and every other tree has at least two leaves. This means that each of T1 and T2 has at least one leaf that does not overlap with the edges.

Spanning Trees

Nevertheless, the total weight of the resulting trees is the same, and each such tree is a clamped tree of minimum weight. If every vertex of G is labeled, then the set of labeled edges forms a spanning tree with minimum weight.

FIGURE 1.41. The city graph.

Counting Trees

Note that in the previous example, none of the leaves of the original treeT appear in the sequence. Subscript 2 is the one, so we place an edge between verticesv2andv4, the first subscript in the sequence.

FIGURE 1.45. Labeled trees on three vertices.

Trails, Circuits, Paths, and Cycles

The Bridges of K¨onigsberg

Construction of bridges in K¨onigsberg. six of the seven bridges in K¨onigsberg and these bridges can be identified by consecutive pairs in the sequence: AB, BD, DA, AC, CA, AB. Prove this in the context of Euler letter sequences. a) if landmass Li is connected by 5 bridges, then L will appear 3 times in any representation of a path that crosses all bridges once.

FIGURE 1.51. Layout of the bridges in K¨onigsberg.

Eulerian Trails and Circuits

Surely if an Eulerian circuit exists, then so does an Eulerian route (the circuit is just a closed route). But are there graphs that are not Eulerian but do contain an Eulerian trace. For each of the following, draw an Eulerian graph that satisfies the conditions, or prove that no such graph exists. a) An even number of vertices, an even number of edges.

Fleury's Algorithm for Identifying Euler Circuits Given: An Euler graph with unlabeled edges.

FIGURE 1.54. Are any of these Eulerian?

Hamiltonian Paths and Cycles

The proof of Ore's theorem is similar to the above and is left to you as an exercise. You have already seen this graph — it is the graph you should have obtained as a complement to the line graph in K5 in Exercise 7b of Section 1.1.3. If Sis is a collection of graphs and if G does not contain any of the graphs in Sas-induced subgraphs, then we say that it is GisS-free.

Prove that each of the 18-vertice graphs in Figure 1.65 is 2-connected, claw-free, and untraceable.11.

FIGURE 1.58. The Icosian Game board.

Three Open Problems

The graph in Figure 1.67 is the smallest known graph in which the diversion paths have an empty intersection. There are exactly seven bypass paths, and every vertex of the graph is missed by at least one of these paths. The graph in Figure 1.70 is 3-connected, claw-free and non-Hamiltonian,12 and so the answer to question C-fork=3 is no.

Show that every vertex of the graph in Figure 1.67 is missed by at least one detour in the graph.

FIGURE 1.66. Walther’s example.

Planarity

• Definitions and Examples
• Euler’s Formula and Beyond
• Regular Polyhedra
• Kuratowski’s Theorem

Verify that the graph in Figure 1.70 is actually the line graph of the graph obtained from the Petersen graph by substituting each of the five “spokes”. Proving that a graph is planar is in some cases very simple: all that is required is to display a planar representation of the graph. Flatten the graph in Figure 1.79 so that each of the following areas is the outermost area.

See if you can find an alternative (not necessarily graph-theoretic) proof that there are only five regular polyhedra.

FIGURE 1.72. Original routes.

Colorings

• Definitions
• Bounds on Chromatic Number
• The Four Color Problem
• Chromatic Polynomials

Suppose inG there is no path from w1tow3where all colors on the path are 1 or 3. Suppose inGer there is a path from w1tow3 where all colors on the path are 1 or 3. Determine the chromatic number of the graph of the map of the countries of Africa.

Determine the chromatic number on the graph on the map of the countries of Australia.

FIGURE 1.86. The Birkhoff Diamond.

Matchings

• Definitions
• Hall’s Theorem and SDRs
• The K¨onig–Egerv´ary Theorem
• Perfect Matchings

This collection has an SDR if and only if for every ∈ {1,. k}, the union of any of these sets contains at least elements. Hall's theorem now implies that X can converge to Y if and only if|A| ≤. In other words, the collection of sets has an SDR if and only if for every∈ {1,. k}, the union of any of these sets contains at least elements. The following result about perfect matchings in bipartite graphs is a consequence of Hall's theorem.

Using Tutte's theorem, prove that the graph in Figure 1.119 does not have a perfect match.

FIGURE 1.103. The matching M 1 .

Ramsey Theory

Classical Ramsey Numbers

If you only have one vertex, no matter how you color the edges (ha-ha) of K1, you will always end up with a redK1. We need to know the smallest integers such that every 2-coloring of the edges of Kn contains either a redK2 or a blueK4. To show this, we need to demonstrate two things: first, that there exists a 2-coloring of K3 that does not contain a redK2 or a blueK4, and second, that any 2-coloring of the edges of K4 at least contains one of these as a subgraph.

For the second point, suppose that the edges of K4 are 2-colored somehow.

FIGURE 1.120. A 2-coloring of K 5 .

Exact Ramsey Numbers and Bounds

On the other hand, if at least one ofxy,xz,yzi is colored red, we have a redK3. Since S contains at least four vertices, and since R(2,4) = 4, the 2-coloring of the edges that are inside S must produce either a redK2 or a blueK4 in S itself. Since T contains at least six vertices , and since R(3,3) = 6, the 2-coloring of the edges that are inside T must produce either a redK3 or a blueK3 in T itself.

So we proved that any 2-coloring of the edges of a complete graph on 9 vertices (or more) produces either a redK3 or a blue K4.

FIGURE 1.126. A 2-coloring of the edges in K 8 .

Graph Ramsey Theory

Consider the graph consisting of n−1 copies of Km−1, with all possible edges between copies of Km−1. In the following theorem, mK2 denotes the graph consisting of copies of K2, and nK2 has a similar meaning.

FIGURE 1.132. The graph S.

Much more information about Ramsey theory can be found in the book [136] by Graham, Rothschild and Spencer. The book by Barab´asi [17] is a pretty general treatment of graphs (networks) and their relationship to all kinds of phenomena.

Combinatorics

Some Essential Problems

The total number of ways to then order objects is therefore the product of the integers between 1 and . How many ways are there to arrange the order of teams picking in the draft. Calculate the number of ways to deal each of the following five-card hands in poker. a) Even: The values ​​on the cards form a sequence of consecutive integers.

Determine the ratio of the number of possible tickets in Wild Money to the number in Texas Limit in two steps. a).

Binomial Coefficients

The answer is the number of ways to choose four cards from the other 51 cards in the deck, namely 51. We can also count the number of hands that do not include the ace of spades. This is the number of ways to choose five cards from the other 51, that is, 51.

There are 16,920 possible tickets, and in general we see that the number of tickets that exactly match the winning numbers is six.

Multinomial Coefficients

We can use a similar strategy to generate a geometric arrangement of the trinomial coefficients whenm = 3, which we can call Pascal's pyramid. Here the position of each number in plane is shown relative to the positions of the numbers in plane−1, each of which is marked with a triangle. We can use the addition identity to obtain an important generalization of the binomial theorem for multinomial coefficients.

But each arrangement corresponds to an order of the elements of our multiset: the numbers in boxi indicate the positions of objects in the list.

FIGURE 2.1. The first five levels of Pascal’s pyramid.

The Pigeonhole Principle

Is it possible to find an arrangement of the integers from 1 to 10 that simultaneously avoids both an increasing subsequence of length 4 and a decreasing subsequence of length 4? Show an arrangement of the integers between 1 and n2 that has no increasing or decreasing subsequence of length n+ 1. Show an arrangement of the integers between 1 and mn that has no increasing or decreasing subsequence of length m+ 1, and no decreasing subsequence with length n+ 1.

Show that there exists a multiple N ofn whose basic representation is obtained by placing an integer number of copies of the base-b digit with the order 1d2 · · ·dm next to each other.

The Principle of Inclusion and Exclusion

Letting N0 denote the number of beads with none of the three attributes, we then calc. We must name a set and list a number of properties such that the number of elements in the set that do not satisfy any of the properties is φ(n). Exactly three ankle socks have a hole. a) Use Theorem 2.6 to determine the number of black ankle socks without holes.

Using Theorem 2.6, determine the number of students who are enrolled in all four subjects at the same time: mathematics, biology, English, and a foreign language.

FIGURE 2.2. A solution using a Venn diagram.

Generating Functions

• Double Decks
• Counting with Repetition
• Changing Money
• Fibonacci Numbers
• Recurrence Relations
• Catalan Numbers

Let's define it as the number of ways to make turns in change, and let A(x) be a generating function forak:A(x). Therefore, the number of ways to make dimes with either pennies or nickels is given by the generating function. Letnkbe the number of ways to make cents using either pennies or nickels, so its generating function is.

Prove that the number of ways to earn 25kcents in change using only these 66 different coins is.

TABLE 2.2. Computing the number of ways to make k cents in change.