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The theorem on planar graphs

The theorem on planar graphs

HISTORIA MATHEMATICA 12(19X5). 356-368 NOTE The Theorem on Planar Graphs JOHN W. KENNEDY Mathematics Department, Pace AND LOUIS V. QUINTAS Univer...

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NOTE The Theorem on Planar Graphs JOHN W. KENNEDY Mathematics





Pace Plaza,






AND MACEJ Institute of Computer ul. Prz,esyckiego


Science, University 20, 511.51 Wroclow,

of Wroc+aw, Poland

In the late 1920s several mathematicians were on the verge of discovering a theorem for characterizing planar graphs. The proof of such a theorem was published in 1930 by Kazimierz Kuratowski, and soon thereafter the theorem was referred to as the Kuratowski Theorem. It has since become the most frequently cited result in graph theory. Recently, the name of Pontryagin has been coupled with that of Kuratowski when identifying this result. The events related to this development are examined with the object of determining to whom and in what proportion the credit should be given for the discovery of this theorem. 0 1985 AcademicPress.Inc Pendant les 1920s avancts quelques mathkmaticiens &aient B la veille de dCcouvrir un theor&me pour caracttriser les graphes planaires. La preuve d’un tel th6or&me a et6 publiCe en 1930 par Kazimierz Kuratowski, et t6t apres cela, on a appele ce thCor&me le ThCortime de Kuratowski. Depuis ce temps, c’est le rtsultat le plus citC de la thCorie des graphes. Rtcemment le nom de Pontryagin a &e coup16 avec celui de Kuratowski en parlant de ce r&&at. Nous examinons les CvCnements relatifs g ce dCveloppement pour determiner 2 qui et dans quelle proportion on devrait attribuer le mtrite de la dCcouverte de ce th6oreme. 61 1985AcademtcPress,Inc. W kolicu lat dwudziestych kilku matematykbw byto blisko sformubwania twierdzenia charakteryzujpcego graf planarny. Dow6d takiego twierdzenia opublikowany zostaf w roku 1930 przez Kazimierza Kuratowskiego i wkr6tce potem twierdzenie to zaczeto nazywaC twierdzeniem Kuratowskiego. Stat0 sic ono najcz&ciej cytowanym rezultatem w teorii graf6w. Obecnie, identyfikujac to twierdzenie, do nazwiska Kuratowskiego dodaje sic nazwisko Pontriagin. Badamy prowadzqcy do tego rozwdj wydarzeii, aby okreSIiC komu i w jakim stopniu powinno sic przpisak zasiuge sformu+owania tego twierdzenia. 0 1985Academic Press,Inc.

INTRODUCTION In 1929 Kazimierz


announced, and in 1930 published,

a proof of the

Theorem on Planar Graphs: A gruph is planar K,,, [Kuratowskl

if and only if it does not contain

This exquisite and fundamental graph theory [Berman 19771.

theorem 356

0315-0860/&X5 $3.00 Copyright6 1985 by Academic All rights of reproduction

a subgraph


to either

Kj or

1929, 19301

Press. Inc. in any form reserved.

is the most frequently

cited result in

HM 12



A proof of this theorem was announced independently and at about the same time by 0. Frink and P. A. Smith [1930]. However, since Kuratowski’s paper [1930] was already in press, the Frink-Smith paper which contained this result was not published. Kuratowski wrote [1930, p. 272, footnote 51 that P. S. Aleksandl.ov had told him that L. S. Pontryagin had also proved, but not published, a similar theorem. As a result of its publication [Kuratowski 19301, the Theorem on Planar Graphs br:came known as Kuratowski’s Theorem [Wagner 19371. Recently, however, the name of Pontryagin has been coupled with that of Kuratowski when identifying tliis theorem (see, for example, [Burstein 1978, Kelmans 1978b1). Since the assignment of appropriate credit for mathematical discovery is often a subtle process, the object of this article is to trace and document events related to the origins and niming of this theorem. Neither the variations nor generalisations of this theorem, n)r the different proofs obtained since its first publication by Kuratowski [1930] q-e considered here; these proofs have been amply discussed by C. Thomassen [] 9811. This “contempory history” concerning the Theorem on Planar Graphs highlights the difficulties faced by historians of mathematics in general, since in most cases the history of mathematics deals with events and sources from periods frn-ther in the past than we have had to consider. Before dealing with the history of the naming of the Theorem on Planar Graphs, it is useful to understand its significance. Abstractly, a graph G = (V,E) is defined as a set V together with a set E of twoe.ement subsets of V. Although simple, this concept has spawned a vast number af applications (see, for example, [Bondy & Murty 1976, Roberts 19761). Many of these applications take place in a geometric or topological setting, and it is in this cantext that the Theorem on Planar Graphs plays its role. Specifically, a topologic al interpretation of a graph is obtained if the elements of V are viewed as distinct points in space, and each pair of these points form the endpoints of a simple curve (I’ordan arc) if and only if the pair of points is an element of E. The resulting configuration is a topological representation of the graph G = (V,E). If none of the s.mple curves in such a representation intersect, then this configuration is called an embedding of G. It is clear that a representation of G is not unique. It is known that, in Euclidean 3-space, every graph has an embedding. However, in 2-space, not every graph can be embedded. In particular, not every graph can be drawn in a plane without lines crossing. A graph is called planar if it can be embedded in a plane. Two examples of nonplanar graphs are shown in Fig. 1. The same graphs,



The complete graph on 5 points

The complete bipartite graph on 3. 3 points






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shown in Fig. 2, are drawn in a plane each with only one crossing. The elegance of the Theorem on Planar Graphs is now evident: the Theorem states that a graph can be drawn in a plane without crossings if and only if it does not contain a homeomorph of either of the two graphs shown here. CIRCA


The principal results contained in [Kuratowski 19301 were announced at a meeting of the Warsaw Section of the Polish Mathematical Society on June 21, 1929 [Kuratowski 1929; 1930, 271, footnote I]. The date cited in [Kuratowski 19291 is May 31, 1929; however, it appears that the records of the Polish Mathematical Society are probably in error, and the date of June 21 cited in [Kuratowski 1930, 27 1, footnote l] is correct. In another footnote to his paper Kuratowski states: I have learned from Mr. Alexandroff, that a theorem for graphs, analogous to my theorem, had been found by Mr. Pontjagin several years ago, but has not been published so far. [Kuratowski 1930, 272, footnote 51

We are not able to contact P. S. Aleksandrov prior to November 16, 1982, the date of his death [Notices of the American Mathematical Society 19831. We were successful in corresponding with L. S. Pontryagin. In response to our initial enquiry [Letter to Pontryagin, 19821, Pontryagin tells us that in the winter of 1927-1928, while a second-year student at the Moscow State University, he proved the Theorem on Planar Graphs. In his own English translation, the circumstances were as follows: The head of our seminar P. S. Aleksandrov formulated Kuratowski’s theorem on nonplanar graphs, more precisely the theorem saying why some graphs cannot be homeomorphitally transform[ed] to the plane. Trying to reconstruct the proof of this theorem 1 [found] that the theorem given by P. S. Aleksandrov [was] wrong and [I found] the correct theorem. [Letter from L. S. Pontryagin, 19821

After stating the correct theorem

he goes on to say,

The theorem given by P. S. Aleksandrov as Kuratowski’s theorem said that any graph which can not be homeomorphically transform[ed] to the plane, contains [a subgraph], which is homeomorphic to the graph K3,, (maybe in Kuratowski’s theorem [it] said the graph KS, I don’t remember it [now]). Hence this theorem was wrong. [Letter from L. S. Pontryagin, 19821

Thus, according to Pontryagin the theorem credited to Kuratowski, as relayed to him by Aleksandrov, was wrong since only one of the two forbidden subgraphs was specified as the necessary condition.

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It is difficult to determine the exact details of these events since Aleksandrov spent the academic year 1927-1928 with H. Hopf in Princeton, where they laid the plans for their classic book Topologie [Aleksandrov & Hopf 19351. No recorded der.ails of communications between Aleksandrov and Pontryagin have as yet been found. Aleksandrov’s version of the theorem could have been communicated to Pontryagin by letter or verbally during a visit to Moscow. It is also difficult to de:ermine exactly when and how Aleksandrov learned of Kuratowski’s result; celtainly, a verbal version could have been distorted along the way. In any case, Pontryagin’s letter concludes: My result was not submitted for publication by P. S. Aleksandrov because he wanted [me to] prove the same result for one-dimensional locally-connected [continua]. [Letter from L. S. Pontryagin, 1982]

We asked Pontryagin [Letter to Pontryagin, 19831 to clarify his comment that Kuratowski had made a mistake in the statement of the theorem (as relayed by Aleksandrov) and requested copies of related papers (or references to them) on thi: Theorem on Planar Graphs or the similar result for one-dimensional locally cannected continua. His reply in English, in its entirety, was as follows: Thank you very much for your letter of 17 January, 1983. I have no published paper on nonplanar graphs. But I remember exactly that I [found] the mistake in Kuratowski’s theorem and told P. S. Aleksandrov about it in Winter 1927-28. Maybe P. S. Aleksandrov told Kuratowski about it, but I don’t know it. Maybe Kuratowski himself [found] the mistake. That’s all the information I can tell you [Letter from L. S. Pontryagin, 19831

ln 1973, Kuratowski [1976] recalled circumstances relating to “the situation in topology in the early 20th-or let us say-in the first quarter of this century” and dilcussed, in particular, “the point-set topology, started by Cantor and developed further by Frechet, and Hausdorff, and later by the new-born Polish and Russian topological schools.” He concluded: No wonder that we became interested in the problem of the characterisation of graphs lying in the plane. This problem was: are there typical graphs which have to be contained in a graph in order that this graph cannot be topologically embedded in the plane? I must confess that when I started to think that problem over, I had in mind just one graph. Namely, the graph called now commonly Ks (according to [Harary 19691). But I noticed soon that there is another one which is also irreducibly non-embeddable in the plane. Namely the graph K3.3. Now (fortunately for me) [there does] not exist any other irreducible skew graph. I proved [this] in 1929 in my paper [Kuratowski 19301. As a matter of fact, I proved slightly more. Namely, I have shown that if .Y is a locally connected continuum which does not contain an infinity of simple closed curves, and if X cannot be embedded in the plane, then X contains either K5 or K3,3. My proof, based on some properties of curves lying in the plane was purely point-set theoretic. Since that time quite a number of proofs based on various ideas, some of them very ingenious (compare [Berge 1958, Dirac & Schuster 1954, Harary & Tutte 1965. Whitney 19331) have been published. [Kuratowski 19761

It is clear from these recollections that Kuratowski had indeed initially formulated the theorem with only one subgraph KS that induced nonplanarity. It is





possible that this early thought was passed by him to Aleksandrov and hence to Pontryagin. However, it is also probable that by the time Kuratowski announced his result in Warsaw he had himself found the other subgraph Kj,j required to complete the theorem, for he certainly published the complete version. There is no evidence to indicate that Pontryagin communicated the addition of K3.3 to Kuratowski prior to publication of the theorem. We asked 0. Frink [Letter to 0. Frink. 19811 what he knew about Pontryagin’s role in the proof of the Theorem on Planar Graphs. Frink replied: I never heard that Pontryagin had a proof of the planar graph theorem prior to 1930, or at that time. Perhaps he did, but I know nothing about it. Karl Menger was supposed to have known the result before 1930, but I have no proof of this. [Letter from 0. Frink, 19811

In 1930 K. Menger did, indeed, publish a paper on the planarity of cubic graphs. Not only did he work in this area, but he was in a position to conjecture such a result for graphs in general. Menger’s own recollection on exactly how he was led to his theorem is as follows: Prior to [1926], I had only one graph-theoretical experience. In 1924 I had studied in Reidemeister’s Vienna seminar, the theorem of Tait about normal maps, i.e., maps with no point belonging to more than three countries and in which the combined frontiers, therefore, correspond to cubic graphs. According to this theorem, a normal map can be colored with four colors if and only if three colors are sufficient to color the edges of its corresponding cubic graph. But on all surfaces of higher genus there exist normal maps which require more than four colors . _ _ and, consequently. cubic graphs that require more than three. Only in the plane (or equivalently on the sphere) was there a chance that three colors might be sufficient for all cubic graphs. First of all, I therefore asked for an intrinsic characterization of planar cubic graphs. But, to my surprise, the literature on graphs (extensive even in 1925) did not provide an answer to my question. So I went ahead and formulated it myself. I realized that each cubic graph that is skew (nonplanar) must contain one that is irreducibly skew (i.e., skew though all of its proper subgraphs are planar); and I further observed and proved by induction according to the number of topological circles in the graph, that all skew cubic graphs in fact contain one and the same irreducibly skew configuration, the edges of a tetrahedron with two points on opposite edges connected by an edge that is otherwise disjoint from the tetrahedron or. in other words, the well-known configuration of three houses and three utilities. But since, by an irony of the logical situation, this skew configuration is among those that can be colored with three colors, I did not further pursue the color problem. While at that time I mentioned my result to several topologists, including Brouwer, Hurewicz, and Aleksandrov, 1 did not publish it except as a short note [Menger 19301. the year when Kuratowski’s fundamental paper [1930] appeared and in which he characterized all skew graphs (not only the cubic skew graphs). Kuratowski’s widely known characterization is by the fact that each skew graph contains at least one of two irreducibly skew configurations: either the graph described above or the complete graph on five vertices. Busy with the development of general curve theory [Menger 1927, 1932, Urysohn 19271, I returned to graphs only in connection with a problem raised by the theory. 1Menger 19811

We also asked Frink to confirm a quotation ascribed to him in 1976 related to the timing of the Frink-Smith announcement [1930] and the submission of their paper to the Transactions of the American Mathematical Society:

Hhl 12



Unfortunately Kuratowski’s proof came out in Fundamenta [Mathematical just at that time, and equally unfortunate was the fact that our proof was similar to Kuratowski’s. Hence our paper was simply rejected by the Transactions. [Briggs, Lloyd. & Wilson 1976, 148)

Fr nk confirmed


The quote from my letter to Dr. Wilson is accurate. Kuratowski’s proof was actually different from ours, since he did not use the notion of an irreducible non-planar graph, but the two papers were not different enough so that ours could be published. [Letter from 0. Frink, 19811

Furthermore, noted that:

in an address on independent

discoveries in graph theory, F. Harary

One of the most fundamental theorems [in graph theory] is that of Kazimierz Kuratowski characterizing planar graphs [Kuratowski 19301. When it appeared in 1930, two American mathematicians Orrin Frink and Paul A. Smith, had already submitted (independently, and independently of each other) papers containing precisely the same theorem, which they promptly withdrew. [Harary 19791

The statement about Frink and Smith submitting “papers”--“independently, ani independently of each other”-appears to be an error of transcription, since Hilrary later writes: For [the Theorem] was independently found and proved by Frink and Smith [1930] who even sent an abstract of the result to the American Mathematical Society in 1930. [Harary 1981, 2181

This is further clarified by Frink, commenting

on the former assertion:

Harary is quite wrong in saying that Paul Smith and I worked independently, We knew each other well, having been together for a year or more at Princeton. I was then interested in graph theory, having published a proof of Petersen’s Theorem in the Annals in 1926 (see [Frink 19261). We discussed the question, conjectured the result, and then tried (independently perhaps), to prove it. Paul was then at Barnard in N.Y., and I was at Penn State. We corresponded. We were lucky that an abstract of the paper was printed in 1930. It seems to me that almost anybody who had thought of the question would soon conjecture the answer. Poor Paul Smith is no longer with us. [Letter from 0. Frink, 19811

..n a letter from 0. Frink to M. Askew sent to us by R. J. Wilson, Frink informs us that: Unfortunately, I no longer have any manuscripts relating to the work which Paul A. Smith and I did about 1930 on irreducible non-planar graphs. [Letter from 0. Frink to M. Askew, 1976 (1974?)]

K. Borsuk, a very close friend of Kuratowski,

has these comments:

I was told by Professor R. Engelking, who talked with Professor Kuratowski about it, that after the theorem was proved, Professor [Kuratowski] mentioned this result in a letter to Professor Alexandroff. Professor Alexandroff answered that the same result (or similar) had been already obtained by Pontryagin, who presented it during his seminar. As far as I know the result of Pontryagin has never been published so that to give him credit is problematic. [Conversation with K. Borsuk, 19811



M. Burstein



also recalls

. . . that this fact [that the result was obtained by Pontryagin in 1927 but not published] was confirmed by my teacher of topology Prof. G. Chogoshvili, who was present at the meeting of the MOSCOW Mathematical Society in 1927; he heard L. S. Pontryagin reporting the result without proof. [Letter from M. Burstein, 1981)

R. Engelking, in a letter which we translate summarizes his version of events as follows:

(from its Polish) in its entirety,

Thank you for your letter of August 5 [ 19831 concerning the Kuratowski theorem on planar graphs. The following is what I can say about this matter. 1. Professor Kuratowski gave a talk about his result during the PTM [Polskie Towarzystwo Matematyczne] seminar on May 31. 1929 [see [Kuratowski 19291 and earlier comment concerning this date]-the note was published in 1930. 2. Professor Kuratowski himself told me he corresponded with Alexandroff, but not with Pontryagin. I doubt if any letter from Alexandroff [to Kuratowski] survived since the archive [of Kuratowski] was lost during the second world war. In the inheritance [of Kuratowski], which passed through my hands (and now belongs to the Polish Academy of Sciences Museum, Paiac Staszica, Warszawa), I recall only one pre-war letter, from FrCchet. It is likely that the letters to Alexandroff [from Kuratowski] survive and you could ask Professor Ju. M. Smimov of the Mowcow National University about this. 3. From conversations with Professor Kuratowski I remember the following sequence of events: (a) Kuratowski proved the theorem and (b) informed Alexandroff about it (c) Alexandroff responded that Pontryagin had obtained a similar result (d) Kuratowski referred in a later paper [Kuratowski 19301 to Pontryagin but (e) Pontryagin did not publish his proof. (It seems to me that it is not clear what Pontryagin had really proved; perhaps a theorem for graphs. Kuratowski had [proved the theorem] for local dendrites.) 4. In recent years, while talking with Professor Kuratowski, I had the feeling that he thought he had been too magnanimous to Pontryagin [in his footnote [Kuratowski 1930, p. 27211. [Letter from R. Engelking. 19831

With respect to Engelking’s suggestion, efforts to contact Professor Ju. M. Smirnov were unsuccessful. Furthermore, Professor A. P. Yushkevitch [Letter from A. P. Yushkevitch, 19821 of the Institute of the History of Science and Technology, Moscow, was unable to shed any light on Pontryagin’s role concerning the Theorem on Planar Graphs. However, he did mention a paper [Frank1 & Pontryagin 19291 that clearly indicates Pontryagin’s interest in topological problems similar to those being studied by Kuratowski. Nevertheless, as Pontryagin himself said [Letter from Pontryagin, 19831, he did not publish any paper on the specific problem of graph planarity. Thus, prior to 1927 no claims had been made either by Kuratowski or by Pontryagin concerning the Theorem on Planar Graphs. In 1929 Kuratowski announced the complete theorem to the Polish Mathematical Society in Warsaw, and this was subsequently published in 1930 [Kuratoswki 1929, 19301 Due to the loss of Kuratowski’s correspondence during World War II, the fact that Aleksandrov was in the United States, and our inability to obtain correspondence from

Hh[ 12



MWCOW, the period 1927-1929 is shrouded in mystery. Kuratowski’s footnote [ 1930,272, footnote 51 indicates for certain only that he had learned from Aleksandrov of Pontryagin’s interest in the problem. As Kuratowski himself later recollet ted [Kuratowski 19761 he initially proposed a single graph (KS) to characterize noilplanarity, though he added the other (K3,J soon after, a recollection that agt’ees with that of Pontryagin. No additional information concerning these matters is to be found in Kuratowski’s memoirs related to the Polish mathematical scc:ne at that time [Kuratowski 19801. It is possible that the version of Kuratowski’s theorem that Aleksandrov carried back to Pontryagin was this earlier and inComplete version. Nevertheless, there can be no dispute that Kuratowski published the first written and correct proof of the Theorem. It is only much later that the name Pontryagin was coupled with that of Kuratowski in this connection. POST- 1930 13ngelking has suggested [Letter from Engelking, 19831 that Kuratowski came to reeret his footnote mentioning Pontryagin’s interest in the Theorem on Planar Gnkphs because Pontryagin had not published a proof. How is it then that recently the association of Pontryagin with the Theorem has become more widespread? j’ontryagin’s name does not appear in the section on embedding problems for grqphs in the comprehensive technical survey of all Soviet publications in graph thdory up to 1968 [Turner & Kautz 1968, also 19701. In particular, the authors cite the survey of worldwide research in graph theory through 1962, with an emphasis on Soviet contributions, written by A. A. Zykov [1962]. Specifically, The first Soviet paper dealing in part with graph theory was by Kudryavtsev [1948] in 1948, and the first Soviet paper devoted entirely to graph theory was written by Zykov [1949] in 1949. In the years to follow, Soviet capability in graph theory can be attributed most directly to one individual, Zykov. Not only is his own work of high quality, but many of the best papers conclude with an acknowledgment of Zykov’s assistance to the author. {Turner & Kautz 1968, 51

It s further noted that Several Soviet papers have referred to a forthcoming book, Tlw Tllcw,lv of Finite Grctphs. by A. A. Zykov [see [Zykov 196911.We recommend that this book be seriously considered for translation into English as soon as possible after it becomes available in the West. [Turner & Kautz 1968. p. 81

‘The Pontryagin and Kuratowski conditions for planarity are mentioned Burstein, whose paper [Burstein 19781 was the first to draw our attention question considered here. In a subsequent letter, Burstein writes:

by M. to the

As far as I know, the name of L. Pontriagin was mentioned in connection with the Planar Graphs Theorem for the first time in a book Theory of Finite Graphs by Professor A. A. Zykov, Nauka, Novosibirsk, 1969. I am not sure whether this book has been translated into English, but it is a most popular graph theory text book in the USSR. In the chapter devoted to planar graphs it states that the result was obtained by L. S. Pontriagin in 1927, but was never published, and in many papers published in the USSR after this book, the theorem is referred to as the “Pontriagin-Kuratowski Theorem.” [Letter from M. Burstein, 19811



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In fact, the earliest published association of Pontryagin’s name with the Theorem is not in the book by Zykov [ 19691, but in the Russian translation by Zykou of the book by C. Berge [1958] in 1962, to which the following footnote was added: This theorem was introduced (but not published) by L. S. Pontryagin in 1927 and in 1930. and independently of him [Pontryagin], proved again by Kuratowski [1930]. As a result we call it the Pontryagin-Kuratowski Theorem. -translator’s comment. [Zykov translation of [Berge 195X]]


in his own book Zykov states:

[This theorem] has been proved (but not published) in 1927 by L. S. Pontryagin, and then independently obtained by K. Kuratowski [1930]. [Zykov 1969. 4371

Before Zykov’s translation of [Berge 19581, Soviet mathematicians universally referred to the Theorem as the Kuratowski Theorem (see, for example, [Wagner 19371). However, since then and, as Burstein notes, more so since the appearance of Zykov’s book [ 19691, this situation has changed. The Russian translations of [Harary 19691 and [Wilson 19721 in 1973 and 1977, respectively, provide further examples. Both of these translations are edited by G. P. Gavrilov and appeared after the publication of Zykov’s book [1969]. In the translation of [Harary 19691, the frontispiece remains unaltered and contains the original dedication to Kazimierz Kuratowski for his discovery of the significance of KS and K3,3 to conditions for nonplanarity of graphs. However, from page 126 et seq. the translator has altered the text to refer to the Pontryagin-Kuratowski Theorem, adding a footnote explaining: L. S. Pontryagin proved (however, did not publish) the planarity criterion in 1927. Kuratowski (independently of Pontryagin) obtained this result in 1930. This is the reason why we call it the Pontryagin-Kuratowski Theorem. -translator’s comment. [Translation of [Harary 19691, p. 1261

Similarly, in the editorial preface to the translation of [Wilson 19721, Gavrilov refers to the “Kuratowski-Pontryagin Theorem.” Throughout the text the trans“Kuratowski Theorem” and it is indexed as lator has preserved the description such [Wilson 19771. However, at two places editorial footnotes have been added to suggest: Precisely, [the theorem should be] the Pontryagin-Kuratowski Theorem since L. S. Pontryagin proved [but did not publish] the theorem in 1927. -editor’s comment. [Translation of [Wilson 19721, pp. 74, 771

Another illustration of the linking of Pontryagin’s name to the Theorem Soviet scholars was brought to our attention by Burstein: It was Prof. A. A. Zykov, who submitted my paper for publication in the [Journal] of Combinatorial Theory (as one of the editors). He insisted that the name of L. S. Pontryagin be added [to the Theorem]. [Letter from M. Burstein. 198l]

S. Schuster, in his review of [Burstein

19781, reacted:

The reviewer is puzzled by the unusual attachment of Pontryagin’s name to this theorem. It is fairly well known that 0. Frink and P. A. Smith obtained a proof almost simultaneously


HM 12



with K. Kuratowski [see [Frink & Smith 193011; yet their names are never attached to the theorem. No reference is given to justify the linking of Pontryagin’s name to the result. [Schuster 19801


C. Thomassen

has remarked


Kuratowski’s Theorem . . was discovered independently by Frink and Smith [ 19301 and Pontryagin [Burstein 19781, and the restriction of Kuratowski’s Theorem to cubic graphs was found independently by Menger [1930. [Thomassen 19811

EIowever, note that the reference attached to Pontryagin here is in fact a reference t,, the above-mentioned Burstein paper in the JOWYZU~of Combinatorial Theory. A. K. Kelmans refers to the “Kuratowski Theorem” in one paper [1978a], vlhile in another [1978b] in the same proceedings he calls it the “KuratowskiI’ontryagin Theorem.” More surprisingly, when this second paper subsequently reappeared in two parts in the Journal ofGraph Theory [Kelmans 1980, 19811 the citation was changed to the “Kuratowski Theorem.” It should be noted that soon after the appearance of Kuratowski’s paper [ 19301, the now famous theorem was universally referred to as the Kuratowski Theorem :.nd that this practice continued up to the publication, in 1962, of Zykov’s translation of [Berge 19581. Since then it has become increasingly frequent for Soviet mathematicians to link the theorem with the name of Pontryagin, apparently only on the basis of the report that Pontryagin obtained a proof that was never published. In recent years this trend has also been followed by some authors outside the USSR. However, it is our view, in general, that credit for a discovery should be given to those who both make the discovery and are confident enough to express it in print. Of course there are circumstances that can modify this position. For exami)le, the subsequent demonstration of incomplete arguments or a flaw in the origila1 proof of a theorem (several examples of this exist in the history of the “FourL’Jolour Theorem” [Stemple 19791; or significant contributions to the discovery by -Ithers who are acknowledged by the original author; or proof of plagiarism by the “original” author; or even the truly independent discovery and publication of the -esult which is not immediately recognized. An interesting example of the latter is :he work of J. H. Redfield and its relation to Polya’s Hauptsatz and Read’s Superposition Theorem [Sheehan 19831. In the case of Pontryagin’s association with the Theorem on Planar Graphs, none of these extenuating factors apply. True, Kuratowski acknowledged that Pontryagin was interested in the problem, but not that he communicated a contribution to its solution. Pontryagin may-or may not-have obtained a proof at about the same time as Kuratowski, but this was never published. To the best of our knowledge, it was not even written down for the purpose of limited distribution. Indeed, Frink and Smith would appear to have a stronger claim to association with the theorem, since at least their announcement of the result appeared in print. However, as we have noted, by the time Frink and Smith tried to publish their result Kuratowski’s publication had appeared; therefore, their manuscript was withdrawn.





In the final analysis we must leave it to the reader to decide to whom and in what proportion the credit should be given for the discovery of the Theorem on Planar Graphs. ACKNOWLEDGMENTS In addition to contacting many individuals. we distributed a call for information on this project entitled “A Plane Request” at a number of conferences and seminars. This notice was also published in C;ruph Throv Net+js/ezter ll(3) (January 1982) and in Historiu Mathemuricu 9 (1982). 487. We express our sincere thanks to all who responded to our enquiries. and in particular to the editors of Graph Theory Newsletter and Historia Mathematics for their help. John Kennedy thanks the K-M Research Group (Dedham. England), where this work was initiated. Louis Quintas acknowledges support for this work from the K-M Research Group (Dedham, England), the Pace University Scholarly Research Committee, and a Dyson College of Arts and Sciences Faculty Research Grant. Maciej Syslo is indebted to the Alexander von Humboldt-Stiftung (Bonn, Germany) for supporting his research.

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Letters and Notes Obituary, P. S. Aleksandrov, 1896-1982, Notices of the Atnericun Muthematical Society 30 (1983). 396-397. Notes from a conversation with K. Borsuk. January 1981. Letter from M. Burstein to authors. December 8, 1981. Letter from R. Engelking to authors, September 9, 1983. Letter from 0. Frink to M. Askew, dated July 27, 1976. (Given to the authors by R. J. Wilson, who believes the letter might have been written in 1974 and that it was in his possession prior to the publication of [Biggs. Lloyd. & Wilson 19761.) Letter to 0. Frink from authors, November 20, 1981. Letter from 0. Frink to authors. November 27, 1981. Letter to L. S. Pontryagin from authors, June 18, 1982. Letter from L. S. Pontryagin to authors, November 29. 1982. Letter to L. S. Pontryagin from authors, January 17, 1983. Letter from L. S. Pontryagin to authors. March 25, 1983. Letter from A. P. Yushkevitch to authors, September I I, 1982.