The mathematical structure known as a graph has the valuable feature of

Transcription

1 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 1 Intoducing Gaphs The mathematical stuctue known as a gaph has the valuable featue of helping us to visualize, to analyze, to genealize a situation o poblem we may encounte and, in many cases, assisting us to undestand it bette and possibly find a solution Let s begin by seeing how this might happen and what these stuctues look like FIRST, FOUR PROBLEMS We begin with fou poblems that have a distinct mathematical flavo Yet any attempt to solve these poblems doesn t appea to use any mathematics you may have peviously encounteed Howeve, all of the poblems can be analyzed and eventually solved with the aid of a elatively new sot of mathematical object and that object is a gaph The gaph we e efeing to is not the kind of gaph you ve seen befoe Fo example, Figue 11 shows the gaph of the function y = sin x Thatis not the kind of gaph we e efeing to The Poblem of the Five Pinces Once upon a time, thee was a kingdom uled by a king who had five sons It was his wish that upon his death, this kingdom should be divided into five egions, one egion fo each son, such that each egion would have a common bounday with each of the othe fou egions Can this be done? Figue 12 illustates an unsuccessful attempt to satisfy the king s wishes Evey two of the five egions, numbeed 1, 2, 3, 4, 5, shae some common bounday, except egions 4 and 5 Fo geneal queies, contact webmaste@pesspincetonedu

2 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 2 Chapte 1 y 1 p p 2 1 p 2 p x Figue 11 Not the sot of gaph we e talking about Figue 12 Attempting to satisfy the king s wishes If the kingdom can be divided into five egions in the manne desied by the king, then something else would have to be tue Place a point in each egion and join two points by a line o cuve if the egions containing these points have a common bounday If A and B ae two adjacent egions in the kingdom and C and D ae two othe adjacent egions, then it s always possible to connect each pai of points by a line in such a way that these two lines don t coss What we have just encounteed is a gaph (ou type of gaph) fo the fist time A gaph G is a collection of points (called vetices) and lines (called edges) whee two vetices ae joined by an edge if they ae elated in some way In paticula, the division of the kingdom into the five egions shown in Figue 12 gives ise to the gaph G shown in Figue 13 In ode to have a solution to the king s wishes, the esulting gaph must have five vetices, evey two joined by an edge Such a gaph is called a complete gaph of ode 5 and expessed as K 5 Futhemoe, Fo geneal queies, contact webmaste@pesspincetonedu

3 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 3 1 G : Figue 13 The gaph epesenting the egions in Figue 12 2 it must be possible to daw K 5 without any of its edges cossing Since thee is no edge joining vetices 4 and 5 in Figue 13, the division of the kingdom into egions shown in Figue 12 does not epesent a solution In Chapte 10 we will visit the Poblem of the Five Pinces again when we will be able to give a complete solution to this poblem The Thee Houses and Thee Utilities Poblem Thee houses ae unde constuction and each house must be povided with connections to each of thee utilities, namely wate, electicity and natual gas Each utility povide needs a diect line fom the utility teminal to each house without passing though anothe povide s teminal o anothe house along the way Futhemoe, all thee utility povides need to buy thei lines at the same depth undegound without any lines cossing Can this be done? Figue 14 shows a failed attempt to solve this poblem, whee the thee houses ae labeled A, B and C Not only can this poblem be looked at in tems of gaphs, but in tems of gaphs this poblem is extemely simila to the Poblem of the Five Pinces We can epesent this situation by a gaph with six vetices, thee epesenting the thee houses A, B and C and thee epesenting the thee utilities wate (W), electicity (E) and natual gas (NG) Two vetices ae joined by an edge when one vetex epesents a house and the othe epesents a utility This gaph then has nine edges This gaph is denoted by K 3,3, indicating that thee ae two sets of thee vetices each whee each vetex in one set is joined to all vetices in the othe set To solve the Thee Houses and Thee Utilities Fo geneal queies, contact webmaste@pesspincetonedu

4 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 4 Chapte 1?? A B C Wate Electicity Natual Gas Figue 14 The Thee Houses and Thee Utilities Poblem A B W? C E NG Figue 15 The gaph epesenting the situation in Figue 14 Poblem, we need to know whethe K 3,3 can be dawn without any edges cossing The attempted solution of the Thee Houses and Thee Utilities Poblem in Figue 14 gives ise to the gaph shown in Figue 15 We will visit the Thee Houses and Thee Utilities Poblem as well in Chapte 10 and explain how to solve the poblem In ou next poblem a gaph will be intoduced whose vetices epesent people Hee we assume evey two people ae fiends o stanges The Thee Fiends o Thee Stanges Poblem What is the smallest numbe of people that must be pesent at a gatheing to be cetain that among them thee ae mutual fiends o thee ae mutual stanges? Fo geneal queies, contact webmaste@pesspincetonedu

5 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 5 b b b b b b b (a) (b) Figue 16 The answe to the Thee Fiends o Thee Stanges Poblem is neithe fou no five Hee too the situation can be epesented by a gaph, in fact by a complete gaph Suppose that fou people ae pesent at a gatheing Then we have a gaph with fou vetices, coesponding to the fou people We join evey two vetices by an edge to indicate that these two people ae fiends o ae stanges, esulting in the complete gaph K 4 with fou vetices and six edges To indicate whethe two people ae fiends o ae stanges, we colo the edge ed () if the two people ae fiends and colo the edge blue (b) if the two people ae stanges Thus thee mutual fiends would be epesented by a ed tiangle in ou gaph and thee mutual stanges would be epesented by a blue tiangle The situation shown in Figue 16a shows that with fou people it is possible to avoid having thee mutual fiends o thee mutual stanges Likewise, when we colo the complete gaph K 5 as in Figue 16b, we see that this situation can even be avoided with five people It tuns out that the answe to the Thee Fiends o Thee Stanges Poblem is six, howeve In fact, we believe that we can convince you of this, even so ealy in ou discussion We state this as a theoem Theoem 11: The answe to the Thee Fiends o Thee Stanges Poblem is six That is, among any six people, thee must be thee mutual fiends o thee mutual stanges Poof: We ve aleady seen that the answe is not five So what we must do is conside the complete gaph K 6 with six vetices whee each edge is coloed ed o blue and show that thee ae thee vetices whee all thee edges joining them have the same colo Fo geneal queies, contact webmaste@pesspincetonedu

6 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 6 Chapte 1 v u w y z w v y z x (a) Figue 17 Poving Theoem 11 u x (b) Let s denote the vetices of K 6 by u,v,w,x, y, z and look at u, say Then thee ae five edges leading fom u to the othe five vetices At least thee of these five edges must be coloed the same, say ed Suppose that thee ed edges lead to v, w and x as shown in Figue 17a It s not impotant what the colos ae of the edges leading u to y and z Thee ae thee edges joining the pais of vetices among v, w and x If even one of these edges is ed say the edge between v and w is ed then u,v and w epesent thee fiends at the gatheing, epesented by the ed tiangle uvw On the othe hand, if no edge joining any two of the vetices v, w and x is ed, then all thee of these edges ae blue, implying that v, w and x ae mutual stanges at the gatheing, epesented by the blue tiangle vwx, which is shown in Figue 17b whee the edges of the blue tiangle vwx ae dawn with dashed lines Although the next poblem is not well known histoically, it is a pactical poblem and shows how gaphs can be used to analyze a poblem that we all might encounte A Job-Huntes Poblem A counselo in a high school has contacted a numbe of business executives she knows fo the pupose of finding summe jobs fo six hadwoking students: Hay, Jack, Ken, Linda, Maueen, Nancy She found six companies, each of which is willing to offe a summe position to a qualified student who is inteested in the business The six business aeas ae achitectue, banking, constuction, design, electonics, financial The Fo geneal queies, contact webmaste@pesspincetonedu

7 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 7 six students apply fo these positions as follows: Hay: Jack: Ken: Linda: Maueen: Nancy: achitectue, banking, constuction; design, electonics, financial; achitectue, banking, constuction, design; achitectue, banking, constuction; design, electonics, financial; achitectue, banking, constuction (a) How can this situation be epesented by a gaph? (b) Can each student obtain a job fo which he o she has applied? SOLUTION: (a) We constuct a gaph G with 12 vetices, 6 of which epesent the 6 students, which we denote by H, J, K, L, M, N (the fist lettes of thei fist names), and the othe 6 vetices epesent the 6 positions a, b, c, d, e, f, epesenting achitectue, banking, constuction, design, electonics, financial An edge joins two vetices if one vetex epesents a business and the othe epesents a student who applied fo a position in that business aea (See Figue 18) (b) Yes The edges Ha, Je, Kd, Lb, Mf, Nc within the gaph G show that this is possible (See Figue 19) In this situation, Ken will have a summe job in the aea of design If this business decides that they would athe hie someone othe than Ken, will all six students still be able to have a summe job fo which they applied? We ll see moe about these kinds of matching poblems in Chapte 7 G : a b c d e f H J K L M N Figue 18 Modeling job applications by means of a gaph Fo geneal queies, contact webmaste@pesspincetonedu

8 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 8 Chapte 1 a b c d e f H J K L M N Figue 19 Illustating the job situation C d g c A e D a B b f Figue 110 A famous poblem concening Königsbeg and its seven bidges NEXT, FOUR FAMOUS PROBLEMS We now look at fou poblems that ae not only impotant in the histoy of gaph theoy (which we will descibe late in the book) but which led to new aeas within gaph theoy In 1736 the city of Königsbeg was located in Pussia (in Euope) The Rive Pegel flowed though the city dividing it into fou land aeas Seven bidges cossed the ive at vaious locations Figue 110 shows a map of Königsbeg whee the fou land egions ae A, B, C, D and the bidgesaea,b,,g The Königsbeg Bidge Poblem Is it possible to walk about Königsbeg cossing each of its seven bidges exactly once? Königsbeg and this poblem can be epesented by a gaph G well, not exactly a gaph Thee ae fou vetices in G, one fo each land egion and two vetices ae joined by a numbe of edges equal to the numbe of bidges joining these two land egions What we get hee is called a multigaph because moe than one edge can join the same Fo geneal queies, contact webmaste@pesspincetonedu

9 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 9 A c a d b C e f B Figue 111 The multigaph epesenting the Königsbeg Bidge Poblem g D pai of vetices This multigaph G is shown in Figue 111 In tems of this multigaph, solving the Königsbeg Bidge Poblem is the same as detemining whethe it is possible to walk about G and use each edge exactly once Actually, thee ae two poblems hee, depending on whethe we ae asking whethe thee is a walk in Königsbeg that ends whee it began o whethe thee is a walk that ends in a land egion diffeent fom the one whee it began A solution to both poblems will be povided in Chapte 5 In 1852 it was obseved that in a map of England, the counties could be coloed with fou colos in such a way that evey two counties shaing a common bounday ae coloed diffeently This led to a much moe geneal poblem The Fou Colo Poblem In a map consisting of egions, can the egions be coloed with fou o fewe colos in such a way that evey two egions shaing a common bounday ae coloed diffeently? The map in Figue 112 is divided into 10 egions These egions ae coloed with fou colos, whee the colos ae 1, 2, 3, 4 It tuns out that the egions of this map cannot be coloed with thee colos so that evey two egions shaing a common bounday ae coloed diffeently, howeve This example, and the Fou Colo Poblem in geneal, can be looked at in tems of gaphs A point is placed in each egion and, like the Poblem of the Five Pinces, two points ae joined by a line if the egions have a common bounday Evey gaph constucted in this way can be dawn without any edges cossing Instead of coloing egions, we can colo the vetices of the esulting gaph so that evey two vetices joined by an Fo geneal queies, contact webmaste@pesspincetonedu

10 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 10 Chapte Figue 112 A map whose egions can be coloed with fou colos Figue 113 Figue 112 A coloing of the vetices of the gaph epesenting the map in edge ae coloed diffeently This is illustated in Figue 113 fo the map in Figue 112 The Fou Colo Poblem will be discussed in moe detail in Chapte 11 In geomety, a polyhedon (theplual is polyheda) is a thee-dimensional solid whee the bounday of each face is a polygon Figue 114 shows two polyheda: the cube and the octahedon It is common to epesent the numbe of vetices of a polyhedon by V, the numbe of edges by E and the numbe of faces by F These numbes fo the cube and the octahedon ae also given in Figue 114 In both cases, V E + F = 2 In 1750 the poblem occued as to whethe V E + F = 2wasa fomula fo evey polyhedon The Polyhedon Poblem Fo a polyhedon with V vetices, E edges and F faces, is V E + F = 2? Evey polyhedon can be epesented by a gaph whose edges do not coss The gaphs coesponding to the cube and the octahedon ae shown in Figue 115 Hee the numbe n of vetices of the gaph is the numbe V of vetices of the polyhedon, the numbe m of edges of the Fo geneal queies, contact webmaste@pesspincetonedu

11 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 11 cube V = 8, E = 12, F = 6 V E + F = = 2 octahedon V = 6, E = 12, F = 8 V E + F = = 2 Figue 114 The cube and octahedon gaph of the cube n = 8, m = 12, = 6 n m + = = 2 gaph of the octahedon n = 6, m = 12, = 8 n m + = = 2 Figue 115 The gaphs of the cube and octahedon gaph is the numbe E of edges of the polyhedon and the numbe of egions of the gaph (including the outside egion) is the numbe F of faces of the polyhedon If it could be shown that n m + = 2foall such gaphs, then the Polyhedon Poblem would be solved This too will be discussed in Chapte 10 Anothe polyhedon is the dodecahedon, shown in Figue 116 Fo this polyhedon, V = 20, E = 30 and F = 12 and, once again, V E + F = 2 It was obseved in 1856 that a ound-tip can be made along edges of the dodecahedon passing though each vetex exactly once Detemining such a ound-tip is known as an Aound the Wold Poblem Consequently, a ound-tip can be made along edges of the gaph of the dodecahedon passing though each vetex exactly once The gaph coesponding to the dodecahedon is shown in Figue 117 whee a tip aound the wold on this gaph can be found by following Fo geneal queies, contact webmaste@pesspincetonedu

12 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 12 Chapte 1 Figue 116 A dodecahedon Figue 117 Aound the wold on the gaph of the dodecahedon the edges dawn in bold The question occus then as to which gaphs have this popety The Aound the Wold Poblem Which gaphs have the popety that thee is a ound-tip along edges of the gaph that passes though each vetex of the gaph exactly once? This poblem will be discussed in Chapte 6 GRAPHS, GAMES, GALLERIES AND GRIDLOCK The game of chess has always been consideed to be a mathematical game Pehaps then it comes as no supise that thee ae puzzles and poblems involving chess that have connections to gaph theoy The fist of these has been taced back to the yea 840 A knight is a chess piece that can move fom one squae to anothe squae that is two squaes fowad, backwad, left o ight and one squae Fo geneal queies, contact webmaste@pesspincetonedu

13 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs Figue 118 A solution of the Knight s Tou Puzzle pependicula to it A knight theefoe always moves to a squae whose colo is diffeent fom the squae whee it stated The Knight s Tou Puzzle Following the ules of chess, is it possible fo a knight to tou an 8 8 chessboad, visiting each squae exactly once, and etun to the stating squae? Figue 118 shows (1) a chessboad, (2) the solution of the Knight s Tou Puzzle given in 840, whee the numbes on the squaes indicate the ode in which the squaes ae visited and (3) this solution given in tems of a gaph, whee each squae of the chessboad is a vetex and whee two vetices ae joined by an edge if this indicates a move of the knight This poblem theefoe has a geat deal of similaity to the Aound the Wold Poblem mentioned in the peceding section The next chess poblem concens a diffeent chess piece: the queen The queen can move in any diection (hoizontally, vetically o diagonally), any numbe of vacant squaes A queen is said to captue o attack a squae (o a chess piece on the squae) if the queen can each that squae though a single legal move It is known that thee is no way to place fou queens on the squaes of a chessboad so that evey vacant squae can be captued by a queen The following poblem asks whethe this can be done with five The Five Queens Puzzle Canfivequeensbeplacedonan8 8 chessboad so that evey vacant squae can be captued by at least one of these queens? Fo geneal queies, contact webmaste@pesspincetonedu

14 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 14 Chapte 1 Q Q Q Q Q Figue 119 Five queens that can captue evey vacant squae on a chessboad Figue 120 The 12 ooms in an at galley The answe to this poblem is yes One possible placement of five such queens on a chessboad is shown in Figue 119 Once again, let G be the gaph whose 64 vetices ae the squaes of the chessboad, whee an edge joins 2 vetices if a queen can move between these two squaes in a single move The Five Queens Puzzle tells us that this gaph contains 5 vetices such that each of the emaining 59 vetices is joined to at least one of these 5 vetices We will say moe about this when we discuss gaph domination in Chapte 3 Example 12: A famous at galley contains 12 ooms 1, 2,, 12 (see Figue 120) in which expensive paintings ae on display Evey oom has exits leading to neighboing ooms (a) Repesent this situation by a gaph (b) Do thee exist fou ooms whee secuity guads may be placed so that evey oom eithe contains a guad o is a neighboing oom of some oom containing a guad? Fo geneal queies, contact webmaste@pesspincetonedu

15 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs G : (a) (b) Figue 121 Modeling the at galley by means of a gaph SOLUTION: (a) Let G be a gaph of ode 12 whee V ={ 1, 2,, 12 } and two vetices ae adjacent if they epesent neighboing ooms (see Figue 121a) (b) If fou guads ae stationed in ooms 5, 6, 7, 8, then evey oom eithe contains a guad o is a neighboing oom of some oom containing a guad In the gaph G, this means that evey vetex is eithe one of the vetices 5, 6, 7, 8 o is adjacent to one of these (see Figue 121b) This situation may suggest two othe questions: (1) By placing the guads in ooms 5, 6, 7, 8, the eight ooms without guads ae neighboing ooms of exactly one oom with a guad It would be helpful if some of these ooms wee nea to moe than one guad Is it possible to place fou guads in ooms in such a way that the numbe of ooms without guads and which ae neighboing ooms of exactly one oom with a guad is less than eight? (2) Is it possible to place fewe than fou secuity guads in the ooms so that evey oom eithe contains a guad o is a neighboing oom of a oom containing a guad? Example 13: Figue 122 shows an intesection of two steets whee thee is often heavy taffic Thee ae seven taffic lanes L1, L2,,L7 whee vehicles can ente the intesection of these two steets A taffic light is located Fo geneal queies, contact webmaste@pesspincetonedu

16 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 16 Chapte 1 L3 L4 L2 L1 L5 L6 Figue 122 The taffic lanes at an intesection L7 L1 G : L3 L2 L7 L6 L4 Figue 123 Modeling taffic lanes at an intesection by means of a gaph L5 at this intesection Duing a cetain phase of this taffic light, those cas in lanes fo which the light is geen may poceed safely though the intesection (a) Repesent this situation by a gaph (b) Detemine whethe it is possible, with fou phases, fo cas in all lanes to poceed safely though the intesection? SOLUTION: (a) Let G be a gaph with vetex set V ={L1, L2,,L7}, whee two vetices (lanes) ae joined by an edge if vehicles in these two lanes cannot safely ente the intesection at the same time, as thee is the possibility of an accident (See Figue 123) Fo geneal queies, contact webmaste@pesspincetonedu

17 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 17 (b) Since cas in lanes L1 and L2 may poceed safely though the intesection at the same time, the taffic light can be geen fo both lanes at the same time The same is tue fo L3 and L4, fo L5 and L6 and fo L7 We may epesent this as {L1, L2}, {L3, L4}, {L5, L6}, {L7} This can also be accomplished as {L1, L5}, {L2, L6}, {L3}, {L4, L7} The question asked in (b) above suggests anothe question Is it possible fo cas in all lanes to poceed safely though the intesection whee thee ae fewe than fou phases fo the taffic light? Questions of this type will be studied and answeed in Chapte 11 Thee ae occasions when we might choose to give the edges an oientation to indicate a diection o pehaps a pefeence elation Oienting all the edges of the complete gaph K n esults in a tounament with n playes whee the oientation of an edge indicates the winne of a match played between two playes In Chapte 9, we ll discove the following amazing esult: Fo any tounament with n playes, thee is always a way to numbe the playes in such a way that Playe 1 beat Playe 2, Playe 2 beat Playe 3, Playe 3 beat Playe 4 and so up to Playe n 1beat Playe n Figue 124a shows the complete gaph K 5 whee the labels u,v,w,x, y indicate five playes whee evey two will paticipate in some spots match Figue 124b shows the outcome of these 10 matches By efeing to the playes u,v,w,x, y as 2, 3, 5, 1, 4, espectively, we see that Playe 1 beat Playe 2, Playe 2 beat Playe 3 and so on, as shown in Figue 124c y u v y u 2 v 4 3 x (a) w x (b) w 1 5 (c) Figue 124 The outcome of a five-playe tounament Fo geneal queies, contact webmaste@pesspincetonedu

18 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 18 Chapte 1 THE ARRIVAL OF GRAPH THEORY While the games, puzzles, poblems and esults we ve mentioned wee not initially pat of gaph theoy as thee was not yet an aea of mathematics called gaph theoy, all this changed in 1891 when the fist puely theoetical aticle dealing with gaphs as mathematical objects was witten by the Danish mathematician Julius Petesen ( ) That he called these objects by the name gaph was pehaps the deciding facto in that name being used fom that time on While we have used this tem a numbe of times, we have yet to give a fomal definition of a gaph and descibe some of the basic teminology associated with gaphs It is now time to do this Although thee ae some vaiations on how mathematicians define these tems, the definitions we ae about to pesent ae among the most common As we have noted and the eade will have aleady expeienced, a gaph can be epesented by a diagam, like the one in Figue 125 Associated with a gaph, thee ae two sets, a vetex set V and an edge set E Fo instance, the gaph H in Figue 125 has V ={u,v,w,x, y, z} and E ={uv, ux, uz,vw,vx,vz, xz} In evey gaph, these sets ae finite so they cannot have an infinite numbe of points Also, each element of E consists of two diffeent elements of V, whee the ode does not matte (Fo instance, the edge uv is the same as the edge vu) Mathematicians efe to the elements of E as 2-element subsets of V The fomal definition of a gaph adopted by mathematicians goes as follows A gaph G is a finite nonempty set V of objects called vetices (the singula is vetex) togethe with a set E consisting of 2-element subsets of V Each element of E is called an edge of G ThesetsV and E ae called the vetex set and edge set of G In fact, G is often witten as G = (V, E) Sometimes the vetex set and edge set of a gaph G ae expessed as V (G) u v H : z w y x Figue 125 A gaph Fo geneal queies, contact webmaste@pesspincetonedu

19 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 19 and E(G), espectively, to emphasize that the gaph G is involved The numbe of vetices in a gaph G is called the ode of G and the numbe of edges in G is its size The ode and size of a gaph ae typically denoted by n and m, espectively The ode of the gaph H of Figue 125 is n = 6andits size is m = 7 We often epesent a gaph G by means of a diagam (and efe to the diagam as the gaph), whee each vetex is indicated by a small cicle (and called a vetex) and an edge ab is indicated by placing a staight-line segment o a cuve between the vetices a and b Although the edges ux and vz intesect in the diagam of the gaph H of Figue 125, the point of intesection is not avetexofh Since e = uv is an edge of the gaph H of Figue 125, the vetices u and v ae said to be adjacent and e is said to join the vetices u and v The vetices u and w ae nonadjacent vetices Sinceu and v ae adjacent vetices, they ae neighbos of each othe Because e = uv is an edge of H, the vetex u and the edge e ae incident, asaev and e Sinceuv and vw ae incident with the same vetex v, they ae adjacent edges The edges ux and vz ae not adjacent When a gaph G is given in tems of a diagam and we want to efe to the vetex set of G o discuss paticula vetices in G, it is useful to assign each vetex of G a label In this case, G is called a labeled gaph Onthe othe hand, if thee is no paticula advantage to labeling the vetices of G, then G is an unlabeled gaph The gaph F of Figue 126 is labeled, while the gaph J of Figue 126 is unlabeled A gaph with exactly one vetex is a tivial gaphthus a nontivial gaph has ode at least 2 A gaph with no edges is an empty gaph Anonempty gaph theefoe has at least one edge u t x F : J : w z y v Figue 126 A labeled gaph and an unlabeled gaph Fo geneal queies, contact webmaste@pesspincetonedu

20 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe 20 Chapte 1 THE FIRST THEOREM OF GRAPH THEORY The numbe of edges incident with a vetex v in a gaph G is called the degee of v and denoted by deg G v When the gaph G unde consideation is undestood, we wite the degee of v moe simply as deg v Ifv is a vetex in a gaph G with n vetices, then v can have at most n 1 neighbos; that is, if v is a vetex in a gaph G of ode n, then 0 deg v n 1 In the gaph H of Figue 125, deg y = 0, deg w = 1, deg u = 3, deg x = 3, deg z = 3, deg v = 4 A vetex of degee 0 is an isolated vetex, while a vetex of degee 1 is called an end-vetexthevetexy is theefoe an isolated vetex and w is an endvetex The minimum degee of a vetex of G is denoted by d(g) o simply d while the maximum degee is denoted by (G) o, whee d and ae the lowecase and uppecase Geek lette delta, espectively Fo the gaph H of Figue 125, d = 0and = 4 Notice that when the degees of the vetices of any gaph G ae added, each edge of G is counted twice This gives us the following esult, namely a theoem that is sometimes called the Fist Theoem of Gaph Theoy since it is felt that if anyone wee to study gaph theoy on his o he own, this would likely be the fist esult he o she would discove In any gaph, the sum of the degees of the vetices is twice of the numbe of edges Moe fomally, and einfocing you vocabulay, we state this fact as follows Theoem 14: Let G be a gaph of ode n and size m with vetices v 1,v 2,,v n Then deg v 1 + deg v 2 + +deg v n = 2m So how many edges can a gaph of ode n have? Fo a gaph to have maximum size, it would have to be complete, whee evey vetex is adjacent to all othe vetices and so evey vetex would have degee n 1 Applying Theoem 14 then tells us that (n 1) + (n 1) + +(n 1) = 2m Fo geneal queies, contact webmaste@pesspincetonedu

21 Copyight, Pinceton Univesity Pess No pat of this book may be means without pio witten pemission of the publishe Intoducing Gaphs 21 and theefoe m = n(n 1)/2 Anothe way that we may answe this question is to obseve that fo an edge e, thee ae n choices fo one vetex in e and n 1 choices fo the othe vetex, o n(n 1) choices in all But since vu is the same as uv fo evey edge uv, the numbe n(n 1) counts each edge twice and so m = n(n 1)/2 Some people have efeed to Theoem 14 as the Handshaking Lemma (A lemma is a mathematical esult, usually not of pimay inteest but useful in poving some theoem of geate inteest) Suppose that thee was a gatheing of people, some pais of whom shook hands with each othe, followed by an inquiy as to how many hands each peson shook If all of these numbes wee added, we would aive at an even numbe, namely twice the total numbe of handshakes that took place A vetex is called even o odd accoding to whethe its degee is even o odd The gaph H in Figue 125 then has two even vetices and fou odd vetices That the gaph H has an even numbe of odd vetices is a consequence of Theoem 14 Coollay 15: Evey gaph has an even numbe of odd vetices Poof: Accoding to Theoem 14, when we add the degees of the even vetices and the degees of the odd vetices of any gaph, the esult is always an even numbe Thus the sum of the degees of all odd vetices is even, implying that the gaph must have an even numbe of odd vetices Degees of vetices will be discussed in consideable detail in Chapte 2 Fo geneal queies, contact webmaste@pesspincetonedu