Luận văn toán học ODD-DISTANCE GRAPHS

Introductions to graph theory1.1.2 Graph Coloring Assigning labels to vertices or edges of graphs is a fundamental tool in manyapplications, proofs and algorithms on graphs.. The chromat

VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Hoang Duc Trung

ODD-DISTANCE GRAPHS

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Hanoi - 2012

VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Hoang Duc Trung

ODD-DISTANCE GRAPHS

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Thesis Advisor : Prof.Dr Moshe Rosenfeld

Hanoi - 2012

My first thanks go to my thesis supervisor Prof.Moshe Rosenfeld, who inspired

my development as a student of Mathematics I have enjoyed more Mathematicsafter long discussions and patient working with my thesis supervisor

I am deeply grateful to the Professors in the Faculty of Mathematics, chanics and Informatics , especially Ass.Prof Vu Hoang Linh and Ass.Prof LeMinh Ha , who accepted and encouraged me to work with Prof Moshe Rosenfeld

Me-Last but not least, I would like to thank my math friends in advanced andhonor programs

Hoang Duc Trung

Ha Noi, Novemver, 2012

1.1 Introductions to graph theory 2

1.1.1 Graph basics 2

1.1.2 Graph Coloring 5

1.1.3 Planar graphs 6

1.1.4 The Unit distance graph 8

1.1.5 The odd distance graph 9

1.2 Two four color problems in the plane 10

1.2.1 The original four color problem 10

1.2.2 The alternative four color problem in the plane 11

1.3 The Probabilistic Method 13

1.3.1 Random Graphs 13

1.3.2 Expectation 15

1.3.3 Graph of high girth and high chromatic number 16

2 The integral distance problems 18 2.1 The Erd ¨os-Anning Problems 18

2.2 Counting unit distances 19

3 The odd distance graph 22 3.1 Subgraphs of odd distance graphs 22

3.1.1 2-colorable graphs 22

3.1.2 3-colorable graphs 23

3.1.3 4-colorable graphs 25

3.2 Subgraphs of odd distance graph in higher dimensions 27

3.2.1 Results in 3-dimensional space 27

3.2.2 Results in n-dimensional space 293.3 Chromatic number of the odd distance graph 303.3.1 The chromatic number of Godd(Q2) 32

In this thesis, we study the odd-distance graph and related graphs

The first chapter covers the basic concepts of graph theory We also introducetwo famous infinite graphs: the unit-distance graph and the odd-distance graph

To clarify the motivation, we trace their history, development and related jectures

con-The second chapter introduces the integral distance problem in the plane,that first appeared in the paper of Paul Erdos in 1946 Many other researchersinvestigated many related problems This activity still goes on these days

Chapter 3 covers the odd distance graphs and its investigations from maticians in recent years We try to work not only in the Euclidean two dimen-sional space but also Euclidean n-dimensional space or space of rational points

mathe-C HAPTER 1

Background And Motivation

1.1 Introductions to graph theory

The paper written by Leonhard Euler on the Seven Bridges of Konigsberg andpublished in 1736 is regarded as the first paper in the history of graph theory

Figure 1.1: The seven bridges of Konigsberg

The problem was to find a walk through the city that would cross each bridgeonce and only once The islands could not be reached by any route other thanthe bridges, and every bridge must be crossed exactly once The walk need notstart and end at the same spot Euler proved that the problem has no solution

1.1.1 Graph basics

A graph G is an ordered pair (V(G), E(G)), consisting of a set V(G) of verticesand a set E(G) of edges A pair(u, v)is usually written simply as uv

Graphs are dynamic data structures They can grow (adding vertices and edges)

or shrink (deleting vertices and edges) When a vertex is deleted, all edges dent with it are also deleted For example :

inci-1.1 Introductions to graph theory

Example 1.1 V = {All the students at VNU }

E = {(x,y) — x and y attend the same class}

Example 1.2 V = {All cities in Vietnam }

E = {(x,y) — There is a direct flight from x to y}

We use the term graph for G(V, E)if the edges (x, y) are unordered pairs

If the edges of G(V, E) are ordered pairs (oriented) we call G(V, E) a digraph(directed graph)

The degree of a vertex x∈ V(G) , denoted by dG(v) is the number of edges cent to v in G

adja-A graph G is r-regular if all vertices have degree r

Figure 1.2: A 5-regular graph

A graph G is labelled if the vertices are assigned unique names

Paths,walks

An edge of form(x, x) is called a loop

If a graph G contains the edges(x, y)more than once ,we say that it has paralleledges

A loop-less graph without parallel edges is a simple graph

A walk in a graph is a sequence (v0, E1, v1, E2, , vk − 1, Ek, vk) such that Ei =(vi− 1, vi) The length of a walk is the number of edges it contains

A walk in G(V, E)in which all vertices are distinct is a path

A walk(v0, E1, v1, E2, , vk − 1, Ek, vk)in which v1 =vk is a closed walk

A closed walk in which vi ̸=vj , for i̸= j, 0≤i, j<kis called a cycle

1.1 Introductions to graph theory

Subgraphs, Tree

A graph H(V1, E1) is a subgraph of G(V, E)if V1 ⊂V and E1 ⊂E

A graph H(V1, E1) is an induced subgraph of G(V, E) if(x, y) ∈ E1 if and only if

(x, y) ∈E

A subgraph H(V1, E1) of G(V, E)is a spanning subgraph of G if V1 =V

A graph G =G(V, E) is connected if between any pair of vertces there is a path

A connected graph without cycles is called a tree

Figure 1.3: Example of induced and spanning subgraphs

Figure 1.4: A labeled tree with 6 vertices and 5 edges

1.1 Introductions to graph theory

1.1.2 Graph Coloring

Assigning labels to vertices or edges of graphs is a fundamental tool in manyapplications, proofs and algorithms on graphs We call this procedure coloring

In various applications we use different rules for coloring

Definition The chromatic number of a graph G , denoted by χ(G), is thesmallest number of colors needed to color the vertice so that vertices connected

by an edge are assigned different colors

Example 1.3 The chromatic numbers of Petersen graph is 3.

small-1.1 Introductions to graph theory

Figure 1.7: A proper coloring of the edges of a 4-regular graph by 5 colors

1.1.3 Planar graphs

A planar graph is a graph that can be embedded in the plane, i.e., it can be drawn

on the plane in such a way that its edges intersect only at their endpoints Inother words, it can be drawn in such a way that no edges cross each other Such

a drawing is called a plane graph or planar embedding of the graph Here aresome examples of planar graph

Example 1.5 The Butterfly graph is planar.

Figure 1.8: The Butterfly graph

Example 1.6 The complete graph K4is planar

Figure 1.9: The complete graph K4

We also give some examples of nonplanar graphs :

1.1 Introductions to graph theory

Example 1.7 The complete graph K5is not planar

Figure 1.10: The complete graph K5

Example 1.8 The graph K3,3 is not planar

Figure 1.11: The K3,3 graph

A very famous theorem related to planar graphs is Euler’s theorem It is a nection about number of vertices, edges and faces in a finite, connected planargraph

con-Theorem 1.9 (Euler’s con-Theorem) If a finite, connected, planar graph is drawn

in the plane without any edge intersections, and v is the number of vertices, e

is the number of edges and f is the number of faces (regions bounded by edges,including the outer, infinitely large region), then : v−e+ f =2

Example 1.10 In K4we have v=4, e =6 and f =4

Palanarity being such a fundamental property, the problem of deciding whether

a given graph is planar is clearly of great importance A major step towardsthis goal is provided by the following characterization of planar graphs, due toKuratowski (1930)

Any graph derived from a graph G by a sequence of edge subdivisions is called

a subdivision of G or a G−subdivision Subdivisions of K5and K3,3 are shown inthe following figure

1.1 Introductions to graph theory

Figure 1.12: (a) A subdivision of K5, (b) a subdivision of K3,3

Theorem 1.11 (Kuratowski’s Theorem) A graph is planar if and only if it

con-tains no subdivision of either K5or K3,3

A subdivision of K5 or K3,3 is consequently called a Kuratowski subdivision For

an alternative proof of Kuratowski’s theorem, we can see [17]

1.1.4 The Unit distance graph

In mathematics, and particularly geometric graph theory, a unit distance graph

is a graph formed from a collection of points in the Euclidean plane by connectingtwo points by an edge whenever the distance between the two points is exactlyone Edges of unit distance graphs sometimes cross each other, so they are notalways planar; a unit distance graph without crossings is called a ”matchstickgraph”

Example 1.12 The Petersen graph is a unit distance graph: it can be drawn in

the plane with each edge having unit length

Figure 1.13: Petersen graphOne of the most famous problems in discrete mathematics asks about thechromatic number of unit distance graph , known as Hadwiger-Nelson Problem

1.1 Introductions to graph theory

Problem 1.13 How many colors are needed to color the plane so that no two

points at unit distance are the same color?

The Hadwiger-Nelson problem concerns the chromatic number of unit tance graphs It is known that there exist unit distance graphs requiring fourcolors in any proper coloring, and that all such graphs can be colored with atmost seven colors Surprisingly, the correct value may actually depend on thechoice of axioms for set theory (Shelah and Soifer 2003) (see [12])

dis-Another important open problem concerning unit distance graphs asks howmany edges they can have relative to their number of vertices

1.1.5 The odd distance graph

The odd distance graph Godd is the infinite graph whose vertices are the points

of the Euclidean plane R2, two vertices connected by an edge if their distance

is an odd integer The ”birth” of this graph happened in a conversation betweenM.Rosenfeld and P Erd ¨os in 1994 at conference on Graph Theory, Combinatoricsand Computation in Boca Raton, Florida This graph is interesting since no 4

points in R2 can have all 6 distances odd integers M.Rosenfeld asked : ”what isthe chromatic number of Godd ? ” Erd ¨os added: ”How many distances among npoints in the plane can be odd integers ?” Thus started the pursuit of unveilingthe mysteries of Godd, whose ”close cousin ,” the unit-distance graph, has beenhaunting mathematicians since its introduction in 1950

Here is an example of a subgraph of Godd:

Figure 1.14: A subgraph of Godd

In the above figure , we note that ∠DOA = ∠DOC = 2π3 and ∆AOB, ∆BOCare equi-triangular So, using the laws of cosine, if we put OD=3and OA =5,

1.2 Two four color problems in the plane

we have AD2 =OD2+OA2−2OA.OD.cos∠DOA =32+52−2.3.5.cos120o =49.That gives AD =7

1.2 Two four color problems in the plane

1.2.1 The original four color problem

The conjecture was first proposed in 1852 when Francis Guthrie, while trying tocolor the map of counties of England, noticed that only four different colors wereneeded Francis inquired with Frederick, Francis’s brother, regarding it, whothen took it to De Morgan, mathematics professor in University College Lon-don The problem stated that given any separation of a plane into contiguousregions producing a figure called a map, no more than four colors are required tocolor the regions of the map so that no two adjacent regions have the same color.The following figure indicate a four-coloring of a map of the states of the UnitedStates

Figure 1.15: A four-coloring of United States

In order to translate the Four-Color Problem into the language of graph ory, we need the notion of a face coloring of a plane A k-face coloring of a planegraph is an assignment of k colors to its faces The coloring is proper if no twoadjacent faces are assigned the same color A plane graph is k-face-colorable

the-if it has a proper k-face coloring For example , the figure 1.16 shows a proper4-face- coloring of the triangular prism :

Figure 1.16: A 4-face-coloring of the triangular prism

1.2 Two four color problems in the plane

Conjecture 1.14 (The Four-Color Conjecture) Every plane graph without cut

edges is 4-face-colorable

”The four color theorem was proved in 1976 by Kenneth Appel and WolfgangHaken It was the first major theorem to be proved using a computer Appel andHaken’s approach started by showing that there is a particular set of 1,936 maps,each of which cannot be part of a smallest-size counterexample to the four colortheorem Appel and Haken used a special-purpose computer program to confirmthat each of these maps had this property Additionally, any map (regardless ofwhether it is a counterexample or not) must have a portion that looks like one

of these 1,936 maps Showing this required hundreds of pages of hand analysis.Appel and Haken concluded that no smallest counterexamples exists becauseany must contain, yet not contain, one of these 1,936 maps This contradictionmeans that there are no counterexamples at all and that the theorem is thereforetrue.” (wikipedia)

Figure 1.17: Kenneth Ira Appel and Wolfgang in the 1970s

1.2.2 The alternative four color problem in the plane

In 1950, Edward Nelson , a student at the University of Chicago, formulatedthe alternative four-color problem: What is the minimum number of colors forcoloring the points of the plane so that points at unit distance apart recievedistinct colors Nelson himself showed that at least four colors are needed

Theorem 1.15 The number of colors for coloring the points of the plane so that

points at unit distance apart recieve distinct colors is at least 4

1.2 Two four color problems in the plane

PROOF Place on the given 3-colored plane what we now call The Moser Spindle.Every edge in the spindle has length 1

Figure 1.18: The Moser Spindle

Assume that the seven vertices of the spindle do not contain a monochromaticunit distance segment Call the colors used to color the plane red, white, andblue Let the point A be the red, then B and C must be one white and one blue.Therefore, D is red Similarly E and F must be white and one blue, so G isred We found a monochromatic segment DG of length 1 in contradiction to ourassumption

Soon after learning about the problem form Ed Nelson, John Isbell provedthat the plane can be colored by seven colors The upper bound of seven on thechromatic number follows from the existence of a ”tessellation” of the plane byregular hexagons, with diameter slightly less than one, that can be assignedseven colors in a repeating pattern to form a 7-coloring of the plane

Figure 1.19: A seven-coloring of the plane, and a four-chromatic unit distancegraph in the plane

1.3 The Probabilistic Method

The answer to the alternative four color problem is unknown, but has beennarrowed down to one of the numbers 4, 5, 6 or 7 The correct value may actuallydepend on the choice of axioms for set theory (Shelah and Soifer 2003) (see[12])

1.3 The Probabilistic Method

1.3.1 Random Graphs

A (finite) probabilistic space (Ω, P) consists of a finite set Ω, called the samplespace, and a probability function P : Ω → [0, 1] satisfying ∑ω⊂ ΩP(ω) = 1 Wemay regard the set Gn of all labeled graphs on n vertices (or, equivalently, the set

of all spanning subgraph of Kn) as the sample space of a finite probability space(Gn, P) The result of selecting an element G of this sample space according tothe probability function P is called a random graph

The simplest example of such a probability space arises when all graphs G⊂ Gn

have the same probability of being chosen Because |Gn| = 2N where N = (n2),the probability function in this case is :

P(G) =2−Nfor all G∈ Gn A more refined probability space on the set Gnmay be obtained byfixing a real number p between 0 and 1 and choosing each edge with probability

p, these choices again being independent of one another Because 1−p is theprobability that any particular edge is not choosen, the resulting probabilityfunction P is given by

P(G) = pm(1−p)N−m

for each G ∈ Gn where m = e(G) This space is denoted by Gn,p For example,

G3,p has as sample space the 2(32 )

spanning subgraphs of K3 shown in the ing figure with the probability function indicated

follow-Note that the smaller the value of p, the higher the probability of obtaining asparse graph We are interested in computing or estimating the probability that

a random graph has a particular property

To each graph property, such as connectedness, Hamiltonian, bipartite, nar there corresponds a subset of Gn, namely those members of Gn, whichhave the given property The probability that a random graph has this par-ticular property is then just the sum of the probabiliies of these graph Forinstance, the probability that a random graph in G3,p is connected is equal to3p2(1−p) + p3 = p2(3−2p), the probability that it is bipartite is (1−p)3+

pla-1.3 The Probabilistic Method

Figure 1.20: The probability space G3,p

3(1−p)2p+3(1−p)p2 = (1−p)(1+p+p2), and the probability that it is bothconnected and bipartite is 3p2(1−p)

In a probability space (Ω, P), any subset A of Ω is referred to as an event, andthe probability of the event A is defined by :

P(A) = ∑

ω∈ A

P(ω)

Events A and B in a probability space (Ω, P) are indepenndent if P(A∩B) =

P(A)P(B); otherwise, they are dependent More generally, events Ai, i ∈ I, are(mutually) indepenndent if, for any subset S of I,

P(∩i∈SAi) = ∏

i ∈ S

P(Ai)

For example, if A is the event ’G is connected’ and B is the event ’G is bipartite’

in the space G3,p, then (unless p =0or p=1)

as random variables, because they depend on which graph happens to be lected More generally, a random variable on a probability space (Ω, P) is any

se-1.3 The Probabilistic Method

real-valued function defined on the sample space Ω In the combinatorial text, random variables are frequently integer-valued Here is a typical example

con-Let S be a set of vertices of a random graph G∈ Gn,p We may associate with S arandom variable Xs defined by :

ω ∈Ω : X(ω) =tThe event is denoted for short by X =t Analogously, one may define four relatedevents : X<t, X ≤t, X ≥tand X >t

Random variable Xi, i ∈ I, are (mutually) independent if the events Xi =ti, i∈ I,are independent for all real numbers ti Random variables are dependent if theyare not independent

1.3 The Probabilistic Method

We note that the identity E(XY) = E(X)E(Y)does not hold in general , although

it is valid when X and Y are independent random variables

1.3.3 Graph of high girth and high chromatic number

We return to the notion of a chromatic number χ(G) Observe that if a graphdoes not contain any cycles, X(G) ≤ 2because every component is a tree that can

be colored easily by 2 colors More generally, consider graphs of girth l, whichmeans that the length of the shortest cycle is l If l is large, this means thatstarting from any vertex, the graph looks like a forest within distance l/2+1.One might expect that such graphs can be allso colored using small number ofcolors, since locally they can be colored using 2 colors However, this is far frombeing true, as shown by a classical application of the probabilistic method

Theorem 1.16 For any k and l, there is a graph of chromatic number > k andgirth>l

PROOF We start by generating a random graph Gn,p, where each edge appears

independently with probability p We fix a value λ ⊂ (0,1

l) and we set p =

nλ− 1 Let X be the number of cycles of length at most l in Gn,p The number ofpotential cycles of length j is certainly atmost nj, and each of them appears withprobability pj,therefore :

inequal-Now let us consider the chromatic number of Gn,p Rather than the chromatic

number χ(G) itself, we analyze the independence number α(G) ,i.e the size ofthe largest independent set in G Since every color class forms an independent

set , it’s easy to see that χ(G) ≥ |V(G)|/α(G) We set a = ⌈3plnn⌉ and considerthe event that there is an independent set of size a By the union bound ,we get :

1.3 The Probabilistic Method

Therefore there is a graph where the number of short cycles is X <n/2 and the

independence number α(G) < a We can just delete one vertex from each shortcycle arbitrarily, and we obtain a graph G′ on at least n/2 vertices which has no

cycles of length at most l, and α(G′) < a The chromatic number of this graph is:

C HAPTER 2

The integral distance problems

A planar point set S is called an integral set if all the distances between the ments of S are integers There are several results and conjectures about integralset

ele-2.1 The Erd ¨os-Anning Problems

The first result about integral sets of points was published in 1945 It is the lowing theorem of Anning and Erd ¨os (see [3]) They proved that we can not findinfinite integral set P1, P2, in the plane The first solution of Anning and Erd ¨os

fol-in [3] seem to be complicated A few months later, Erd ¨os gave a simpler proofthat we will consider in the following Interestingly, this proof is rediscovered byTran Nhat Tan, a student in K12-math honor program

Theorem 2.1 If H is an infinite integral set in the plane, then the points of H lie

on one line

PROOF If A, B, C are three points not all on a line and k=max(AB, AC), thenthere are at most 4(k+1)2points P such that PA−PBand PB−PCare integral For |PA−PB| is at most AB and therefore assumes on of the values 0,1, ,k,that is , P lies on one of k+1 hyperbolas Similarly, P lies on one of the k+1hyperbolas determined by B and C These (distinct) hyperbolas intersect in atmost 4(k+1)2points That is a contradiction This completes the proof

We can not have an infinite integral point not all on one line However, wealways can find an integral point set of n points for each given n

Theorem 2.2 For any n we can find n points no three on a line in the plane such

that their distances are all integral