The simple title tells many things. Of course, it could tell almost nothing since a well-known motto says that ‘everything must
have a beginning’. However, another motto is more appropriate here: ‘A journey of a thousand miles must begin with a single
step’. Yes, we remember here the first steps of a journey of more than 60/2 years; steps that set the basis for the first group of
people working on Graph Theory not only at Barcelona, but in the whole of Spain. And all this began when Miguel Angel FiolPeer Reviewe

Yebra, J. L. A.

The simple title tells many things. Of course, it could tell almost nothing since a well-known motto says that ‘everything must

have a beginning’. However, another motto is more appropriate here: ‘A journey of a thousand miles must begin with a single

step’. Yes, we remember here the first steps of a journey of more than 60/2 years; steps that set the basis for the first group of

people working on Graph Theory not only at Barcelona, but in the whole of Spain. And all this began when Miguel Angel FiolPeer Reviewe

UPCommons

The beginnings

UPCommons. Portal del coneixement obert de la UPC

The BeginningsJose´ Luis A. Yebra (invited speaker)Universitat Politcnica de CatalunyaCastelldefelsAbstractThe simple title tells many things. Of course, it could tell almostnothing since a well-known motto says that ‘everything musthave a beginning’. However, another motto is more appropriatehere: ‘A journey of a thousand miles must begin with a singlestep’. Yes, we remember here the ﬁrst steps of a journey of morethan 60/2 years; steps that set the basis for the ﬁrst group ofpeople working on Graph Theory not only at Barcelona, but inthe whole of Spain. And all this began when Miguel Angel Fiol...1 Once ... in 1975-1979I was going to say ”Once upon a time...” but I will be much more precise: Iwill speak of things that happened in the years 1975-79. This means morethan 60/2 years ago. I am sure that some of you cannot remember thisperiod and I see that some others were not yet among us. I think that tobetter understand the beginnings of the Graph Theory group at Barcelonawe should travel back to those years. We need to become acquainted withthe circumstances under which we worked; it is relevant to be conscious ofthe advantages but also the hardships of this period.It was during the 1975-79 years that the Vietnam War ended and theIran revolution took place, while in Spain Franco died and for the ﬁrsttime in the last forty years we enjoyed democratic elections under a newConstitution. In Mathematics, it was the age of public-key cryptographyand the RSA encryption algorithm, as well as the appearance of Mandelbrot3The Beginnings J. L. A. Yebrafractals. More generally the scientiﬁc world celebrated the appearance ofthe ﬁrst personal computer, the ﬁrst CD audio and the ﬁrst human invitro fertilization (which, incidentally, assures the older of us of our naturalconception, perhaps one of the few advantages we have over the youngerones). Of course, there was no cellular phones, no DVD’s and no Internet.These few sentences compose a very fast review of the 1975-79 period andare not enough to understand the limitations encountered. So, I will comeback to this issue.This ﬂash-back is especially important for those under thirty years old,since they have not lived through this period. As the lyrics of La Bohe`me,the Charles Aznavour song goes:La bohe`me Very free translationJe vous parle d’un temps I tell you about a timeque les moins de vingt ans That under-thirtiesne peuvent pas connaˆıtre cannot know aboutMontmartre, en ce temps-la` ... At that time, Miguel Angel ...At that time Miguel Angel was a young student of TelecommunicationEngineering At the UPC and a columnist of the satirical journal “La codor-niz”, surely unaware that new important activities were about to come. Itsorigin should be found in Martin Gardner’s column in Scientiﬁc Ameri-can “Snarks, Boojums and other conjectures related to the four-color-maptheorem” ([2]), together with the providential launch late in 1976 of “Inves-tigacio`n y Ciencia”, the Spanish translation of the magazine, just in time tooﬀer in the April 1977 issue the Snark’s article to a broad Spanish speakingaudience that included Miguel Angel.But let us deviate a little from our theme to pay a small tribute to theimmense ﬁgure of Martin Gardner, who has died just three weeks ago. Hehas been the most important writer in recreational mathematics. Surely allof us have enjoyed many of his writings. When we aﬃrm that without himwe would not be here, we can ask how many other mathematicians shouldalso acknowledge an analogous origin. Thanks a million, Martin, for all thedelightful moments you have given us and for the wonderful world you didopen to us.4The Beginnings J. L. A. Yebra2 From the 4-colour problem to SnarksIn those years there was much research on the so-called 4-colour problem:Can every planar map be coloured using at most four colours in such a waythat regions sharing a common boundary (other than a single point) getdiﬀerent colours? Fallacious proofs were given independently by Kempe(1879) and Tait (1880). Kempe’s proof was accepted until 1890 when Hea-wood showed an error. It is easy to proof that ﬁve colours suﬃce, butreducing the number of colours to four proved to be very diﬃcult. OnApril 1, 1975, Martin Gardner published a map with the claim that it re-quired ﬁve colours if adjacent countries were to receive distinct colours. Ofcourse, the map could be 4-coloured: It was just an April 1 Fool’s joke.To any planar map (where at most three countries meet at any boundarypoint) can be associated a 3-regular planar graph with boundaries betweencountries seen as edges and reciprocally. And it is very easy to proof thatsuch planar map can be 4-coloured if and only if the edges of its associated3-regular graph can be properly 3-coloured. Here properly means that thethree edges meeting at every vertex get diﬀerent colours. As a consequence,it became interesting to ﬁnd 3-regular graphs that could not be properly3-coloured. And if any such graph is planar the 4-colour problem is (nega-tively) solved. The positive result was ﬁnally obtained by Appel and Haken(1977), who constructed a computer-assisted proof that four colours suf-ﬁced. Some mathematicians did not accept it. However, the proof appearedvalid and the result is accepted today as the 4-colour theorem. A shorterproof has since been constructed by Robertson et al. in [4].Martin Gardner’s proposed to use the term snarks for nontrivial 3-regular graphs that are not 3-coloreable. The name comes from LewisCarroll’s ‘The hunting of the Snark’ because of its elusive character. Justa few years before only a handful of snarks were known: The ﬁrst one wasthe ubiquitous Petersen graph on 10 vertices, found in 1898; the next aretwo graphs on 18 vertices due to Blanusa (1946); the following one is agraph on 210 vertices found by B. Tutte, and the last one was a graph on50 vertices found by G. Szekeres in 1973. Then Martin Gardner’s articlepresented the work of Rufus Isaacs who discovered two inﬁnite families ofsnarks in 1975, see [3].As other Martin Gardner articles, it fostered much research on thesubject. Miguel Angel generalized some of Isaacs’s constructions by a quitenew approach based on Boolean Algebra.5The Beginnings J. L. A. Yebra3 Communicating with the scientiﬁc communityOnce the new constructions were ready, a new problem arose: What canman do with them? There was a simple answer: Communicate it to thescientiﬁc community. But remember that we were in 1977. There were nopersonal computers, nor any of those fancy things we can use with them.There was no such thing as e-mail. In those years communication meantto type a letter, to put it inside something called envelope, to stick in thefront side a stamp, to put it in a mailbox and to wait several weeks foran answer. Perhaps I am exaggerating since every body knows what anenvelope and a stamp are, but for how long? The most practical way ofwriting consisted in typing with a writing machine. We should rememberthat, since there were no personal computers, there was no such thing asword processors, style or Grammatik correctors, not even a delete key tocorrect the smallest errors, which were corrected by taking the sheet ofpaper out of the writing machine, creasing it (avoiding uttering four letterwords) and beginning again with a new sheet.Besides, at school people learnt some French, but communicating withthe scientiﬁc community meant to write in English. Therefore, you canunderstand that the ﬁrst two letters that Miguel Angel addressed to MartinGardner were written in Spanish, beginning withBarcelona, 6.XI.77 Barcelona, 22.XI.77Sr. D. Martin Gardner Sr. D. Martin GardnerE.E.U.U. E.E.U.U.Apreciado Sr.: Apreciado Sr.:Soy estudiante ... Como continuacio´n ...Well I do not know what most of us would do when receiving letterswritten in an alien language and related to some relatively aged work. Butwe know what Martin Gardner and Rufus Isaacs did through the answerof the latter, dated December 21, 1977:Dear Sr. FiolMartin Gardner has forwarded me your two letters.Both look very interesting. I regret the delay in re-plying, but this was mainly due to my unsuccessfuleﬀorts to ﬁnd a Spanish translator.6The Beginnings J. L. A. YebraBy I think I understand you because of your beautifuldrawings and because you are travelling a road I havebeen over myself. For example ...Very nice from R. Isaacs but, since he has understood only the drawings,he had missed most of Miguel Angel work. It was imperative to commu-nicate in English. How? Well, look at the way Miguel Angel did it: Firsttype the letter in Spanish, double spaced:Agradezco su carta del 21.XII.77, as´ı como tambie´nel folletoconteniendo su art´ıculo sobre los Snarks. Ambos loshe le´ıdocon mucho intere´s ...Then look for someone that can translate it into English placing thehandwritten translation between the Spanish lines, so as to obtain:Agradezco su carta del 21.XII.77, as´ı como tambie´nel folletoI want to thank you for your letter of December 21,1977 asconteniendo su art´ıculo sobre los Snarks. Ambos loshe le´ıdowell as your paper about the Snarks. I’ve read bothof themcon mucho intere´s ...carefully ...Finally, typeset the English version to achieve some thing likeI want to thank you for your letter of December 21,1977 aswell as your paper about the Snarks. I’ve read bothof themcarefully ...7The Beginnings J. L. A. YebraThe ﬁnal result is a long typeset English letter ... without errors! Andremember: without even a delete key to correct a single error.But this eﬀort pays: The answer from Rufus Isaacs says:Dear Senor Fiol,I owe you my deepest apologies on two grounds.First, ...Second, for under-estimating the depth of your work.I see now you have made contributions of insight andimportance. Congratulations!I am enclosing a paper which I was arranging to pub-lish in a graph theory journal. With your permissionI should like to write a similar piece on your work.(...)I thing you should write a polished paper on your ownabout your ﬁne Boolean logic method of generatingsnarks.(...)And in a letter dated November 29, 1978 Rufus Isaacs oﬀers:I recently spoke to Professor Gore, telling him myhigh opinion of you. He seems interested in your join-ing the faculty...An oﬀer to join the Johns Hopkins University that Miguel Angel couldnot accept because ... he was still a undergraduate student!4 ConclusionUnder my light supervision Miguel Angel wrote his graduation thesis aboutthe construction of snarks using Boolean operators. It is diﬃcult to sum-marize here its more than two hundred pages. I will just restrict myselfto showing below that the Petersen graph is a snark and refer to MiguelAngel’s paper [1] for the way Boolean logic enters the snarks’ world. TheBoolean family contains the BDS family of Isaacs and some later addi-tions by other authors as Loupekhine. But it also contains many more newsnarks.8The Beginnings J. L. A. YebraTo see that the Petersen graph is a snark, notice that to properly colourthe edges of a pentagon, any colour can be used at most twice. Thus, threecolours are needed, using two of them twice and the other one just once.This is the only proper colouring of the pentagon except for permutationof the colours. As a consequence, the edges adjacent to those vertices(we will called them semi-edges) must be coloured as in ﬁgure . Noticethat one colour appears three times in consecutive semi-edges while theother two colours appear just once in the remaining semi-edges. Again thecolouring is unique except for permutations of the colours. When we drawthe pentagon as the ‘inside’ pentagon in the usual representation of thePetersen graph this circular ordering is modiﬁed, so that the three semi-edges equally colored are not consecutive, see Figure 1. As a consequence,the Petersen graph cannot be properly coloured. It is possible to colour atmost three of the ﬁve edges joining the ‘inside’ and the ‘outside’ pentagons,see Figure 1, where the remaining two edges sholud be coloured 1 at oneend and 3 at the other one.~~ 		~~ 	~~~~1232 13233 1323313233113Figure 1: The Petersen graphToday, thirty two years later, all small snarks are known. For instance,there are exactly 2 snarks on 18 vertices, 6 on 20 vertices, 20 on 22 vertices,38 on 24 vertices, 280 on 26 vertices and 2900 on 28 vertices.We could also speak of Miguel Angel’s PhD Thesis (1981); we couldspeak of Minimal Connections, Double Loop Networks, Tessellations, LineDigraphs, the (Δ,D) problem, etc. But this will take us much longer.Today, Miguel Angel is more than sixty year old. This is a consequenceof the simple addition rule. But the most important thing is that insteadof being alone there are more than 60/2 other people working in Combi-natorics and Graph Theory at Barcelona, not counting those people that9The Beginnings J. L. A. Yebrawork at other places after been initiated here.References[1] M.A. Fiol. A Boolean algebra approach to the construction of snarks.Graph Theory, Comb. and Appl. Y. Alavi, G. Chartrand, O.R. Oeller-mann, and A.J. Schwenk., Eds. New York: John Wiley & Sons, Vol.1,493–524, 1991.[2] M. Gardner. Snarks, Boojums and other conjectures related to thefour-colormap theorem. Scientiﬁc American, 234:126–130, 1976.[3] R. Isaacs. Inﬁnite families of nontrivial trivalent graphs which are notTait colorable. Amer. Math. Monthly, 82:221-239, 1975.[4] N. Robertson, D. Sanders, P. Seymour, and R. Thomas. The Four-Colour Theorem. J. Comb. Theory B, 70(1):2–44, 1997.10

The beginnings

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