柯尼希·德奈什Dénes Kőnig

柯尼希·德奈什是著名匈牙利数学家朱利叶斯·科尼格和Eliz 亚历山大·奥本海姆（1853-1916）的儿子。家里有两个男孩；德奈什有一个兄弟György。德奈什在布达佩斯接受了极好的教育，就读于该市最好的学校之一，一所文理中学。这所被称为“Minta”的中学成为匈牙利新型高中的典范。当然，有一位著名数学家作为父亲，德奈什从父亲那里学到了很多数学，但他在中学也有两位优秀的数学老师，即Manó Beke（1862-1946），他于1892-93年在哥廷根师从菲利克斯·克莱因，以及Miklós Szijártó，他曾是厄特沃什·罗兰的学生。德奈什有着极其非凡的高中生涯，因为他取得了对一名学生来说基本上闻所未闻的成就。他的第一篇论文Elementary discussion of two maximum-minimum problems（匈牙利语）于1899年发表，当时他还在上学，还不到十五岁。这相当了不起，但另外少数人也取得了类似的壮举。也许更了不起的是，他在上学期间写了一本61页的书，Mathematical Recreations（匈牙利语）。这本书于1902年出版，即他从布达佩斯“Minta”中学毕业一年后，第二卷于1905年出版，当时他还是大学生。1902年那一卷的序言由Manó Beke撰写，他强调了数学娱乐问题在补充学校课程方面的重要性。他还指出，德奈什不仅包括初等问题，还包括更高级的材料。Beke写道：-

我非常高兴，我的年轻朋友，也是我最亲爱的学生之一，承担了编辑一本名为“数学娱乐”的小册子的工作。他在这方面颇有天赋。他还承担了在学校内外激发对数学问题的兴趣，以及激发普通读者兴趣的任务，让他们处理许多超出高中教学科目的数学问题。这类方向的书籍已经有很多；国外的出版物在这一类型中有最出色的合集。作者从这些合集中选择主题，读者可以在小册子末尾找到所列的来源。即使存在匈牙利语的合集，它也远比这里的作品初级，只包括相当普通的算术和几何谜题。这本小册子不仅包括这类内容，还包括初等问题之外的东西。它不仅包括通常可以用线性方程解决的流行算术谜题（包括谜语），还包括上面提到的在国外印刷的更出色的作品，这些作品引领我们进入奇妙的数字世界。……

Beke和Szijártó都是数学教育改革的有力倡导者，德奈什的这本书显然被视为该计划的一部分。1902年通过中学毕业考试后，他在厄特沃什·罗兰高中数学竞赛中获得第一名。

1902年晚些时候，德奈什进入布达佩斯大学，在那里度过了四个学期。随后他前往德国，在哥廷根大学学习了五个学期，于1906年圣诞节返回布达佩斯。在哥廷根期间，他参加了赫尔曼·闵可夫斯基在1904-05学年关于拓扑学（当时称为Analysis Situs）的讲座。回到布达佩斯后，他由屈尔沙克·约瑟夫指导。1907年，他凭借学位论文Elementary Discussion of Rotations and Finite Rotation Group of a Space of Many Dimensions （匈牙利文）[2]从布达佩斯技术大学获得博士学位：-

德奈什的学位论文处理几何问题，他的导师屈尔沙克·约瑟夫也对数学娱乐和图论感兴趣。屈尔沙克·约瑟夫的著作在德奈什于1902年和1905年出版的数学娱乐书籍中被引用了两处。

获得博士学位后，德奈什加入了布达佩斯技术高等学校的教职员队伍，他的父亲是那里的教授。他最初的职位是演示员，但1908年他成为数学教授的助理。1911年，他与阿尔弗雷德·哈尔发表了两篇联合论文，论文On simply ordered sets（匈牙利文）及其德文译本Über einfach geordnete Mengen Ⓣ（论简单有序集）。到1910年，他已被列为第一助理，次年成为讲师（Docent）。这意味着他可以开设自己的讲座课程，他讲授了Analysis Situs课程。在随后作为讲师的几年里，德奈什开设了Analysis Situs、列线图学、实数、集合论、实数与函数以及图论的课程。尽管直到1927-28学年他才开设名为图论的课程，但从1911年起，他就在其Analysis Situs课程中包含了图论的章节。从1913-14年到1927年，除了教授数学学生外，德奈什还向建筑和化学工程专业的学生教授数学。他一直留在布达佩斯技术高等学校直到去世，并于1935年成为正教授。

在1918年和1920年，德奈什出版了重要的书籍。第一本是The Elements of Analysis Situs which 于1918年出版[6]：-

他以初学者易于接受的方式处理可定向2-流形的拓扑学。这本书不仅是国内，也是国际数学文献中关于这一主题的第一本书。在此之前，这一领域既没有教科书也没有专著（唯一可用的文献是关于波恩哈德·黎曼曲面的）。……几位匈牙利数学家从德奈什的书中学习了曲面的拓扑学。出于对初学者的考虑，德奈什采用了直观的方法。该书证明了主要的分类定理并描述了各种标准形式。表述非常严谨。他经常找到概念和证明的新方法；特别是，他给出了约化为标准形式的简化版本。对于几个证明，他还填补了文献中存在的空白。

第二本书于1920年出版，是Mathematics. Lectures at the Polytechnical University of Budapest for Students in Architecture and Chemical Engineering。

德奈什 在 1904-05 学年 [6] 听过 赫尔曼·闵可夫斯基 关于四色问题的讲座，并受其影响：-

在这些讲座中，除了刻画二维曲面的拓扑性质以及生成各种正规类型之外，赫尔曼·闵可夫斯基 还打算给出 Wemicke 对四色猜想的一个证明。只是在展示该证明的预备步骤时，才发现这个证明是错误的，因此只给出了五色定理的证明。

这些讲座促使 德奈什 对图论日益增长的兴趣，但他对这一主题感兴趣还有其他原因。他曾研究过 大卫·希尔伯特 和 保罗·哥尔丹 关于不变量理论的工作，而他们曾提到 朱利叶斯·佩特森 的论文 Die Theorie der regulären Graphs Ⓣ（正则图论）（1879）。德奈什 看到图论的使用极大地帮助了问题的可视化并有助于解决它们。他决定尝试使图论成为一门体面的数学学科。直到此时，图的使用被视为向儿童展示想法的一种有趣方式，而不是解决深层问题的工具。他在图论方面的第一批出版物，即他开始其使命的地方，是 1911 年发表的两篇匈牙利语论文，即 Graphs on Two-sided Surfaces 和 The Genus of Graphs。1914 年，他参加了在巴黎举行的 Congrès de philosophie mathématique，并在那里作了一场讲座，其中他展示了一个今天被称为“德奈什 定理”的图论结果。这是今天被称为匹配理论的领域中第一个结果。由于第一次世界大战的爆发，包含 德奈什 论文的这次会议论文集直到 1923 年才出版。然而，他在 1916 年的一篇匈牙利语论文中发表了“德奈什 定理”，同年还在德语论文 Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre Ⓣ（论图及其在行列式理论和集合论中的应用）中发表。这项关于二部图因式分解的工作与 菲利浦·霍尔 的婚姻问题密切相关。德奈什 使用图来给出 费迪南德·格奥尔格·弗罗贝尼乌斯 的一个 行列式 结果的更简单证明，似乎导致了两人之间的一些敌意。费迪南德·格奥尔格·弗罗贝尼乌斯 的可约行列式定理于 1912 年发表，而 德奈什 用某个二部图的完美匹配给出的更简单证明，于 1915 年发表在他的论文 Line systems and determinants（匈牙利语）中。

德奈什 的父亲 朱利叶斯·科尼格 于 1913 年去世。他生命中的最后八年一直在研究集合论，特别是 连续统假设。当 Gyula 去世时，他一直在写的一本关于集合论的书几乎完成，但尚未完全完成。德奈什 完成了这本书 Neue Grundlagen der Logik, Arithmetik und Mengenlehre Ⓣ（逻辑、算术和集合论的新基础），在他父亲已经写好的前言中加了一条注记，说他父亲：-

……为[这本书]工作了八年，直到他生命的最后一天。只差几页了。

在感谢 屈尔沙克·约瑟夫 帮助最终完成他父亲的手稿后，德奈什 写道：-

我也向费利克斯·豪斯多夫教授致以诚挚的感谢。他承担了与我一起校对整本书这项痛苦而无回报的工作，从而实现了我父亲的遗愿之一。

然而，对集合论感兴趣的不仅是朱利叶斯·科尼格，德奈什也对该主题做出了一些重要贡献。他对该主题的兴趣始于1908年左右，当时他试图在不假设良序原则的情况下证明费利克斯·伯恩斯坦关于集合等价的两个定理。然而，他对集合论的兴趣并非与他对图论的兴趣无关，因为他经常将他的集合论结果转化为图论术语。他在费利克斯·伯恩斯坦定理上的工作使他给出了今天被称为“德奈什无穷引理”的几种不同版本，即：-

……如果在有限树中路径的长度没有有限上界，那么树中至少存在一条无限路径。

该引理的多种形式出现在德奈什1926年和1927年的一篇论文中，即Über eine Schlussweise aus dem Endlichen ins Unendliche Ⓣ（《从有限到无限的推理》），该文完全致力于无穷引理。Miriam Franchella写道[4]：-

……值得德奈什大加赞扬的是，他能够分离出该引理并理解其广泛的适用性。

德奈什的书Theorie der endlichen und unendlichen Graphen Ⓣ（《有限与无限图论》）于1936年出版，是全球图论兴趣增长的主要因素。它最终被翻译成英文，书名为Theory of finite and infinite graphs（由R McCoart翻译），Birkhauser，1990年。Keith Lloyd在[10]中评论了该英译本：-

德奈什的Ⓣ《有限与无限图理论》（Theorie der endlichen und unendlichen Graphen）最初于1936年在莱比锡出版，是第一部图论教科书。1950年由Chelsea重印，尽管此后出现了许多关于该主题的文本，德奈什的书仍然经常被引用。因此，半个多世纪后才出现英文版，令人惊讶。本书的主体部分由Richard McCoart翻译了全部原始德文文本。……德奈什处理的许多主题几乎在任何关于该主题的文本中都能找到，例如莱昂哈德·欧拉迹、哈密顿圈、迷宫、树、有向图和因子分解，但与许多后来的作者不同，德奈什考虑了无限图和有限图。除了翻译，英文版还包含威廉·托马斯·塔特撰写的43页评论和高洛伊·蒂博尔撰写的德奈什传记素描。在前者中，威廉·托马斯·塔特逐章总结了德奈什处理的材料，并讨论了后来的发展，从而提供了对德奈什工作的历史视角。

我们引用1936年该书引言中的以下内容：-

也许甚至超过人类与自然的相互作用，图论基于人类彼此之间的相互作用。

在其1936年书的序言中，德奈什讨论了图论是拓扑学的一个分支还是组合学的一个分支。他论证是后者：-

……主要因为我们归因于图的元素——顶点和边——完全没有几何内容：顶点是任意可区分的元素，边不过是其两个端点的统一。这种抽象观点——詹姆斯·约瑟夫·西尔维斯特在1873年强调过——将在我们的表示中严格保持，除了一些例子和应用。

至于德奈什的个性，我们引用[6]：-

他是个性格开朗、才华横溢的人。他喜欢与人交往，乐于讲述轶事。凭借讽刺性的幽默，他能极好地取悦同伴。他喜欢他的同事，是数学家们咖啡馆聚会中不可或缺的参与者。

他将永远被视为图论的主要奠基人之一。他常常以这样的话开始讲座：——

图论是数学学科中最有趣的领域之一。

埃尔德什在1977年Journal of Graph Theory第一期上写了一篇“欢迎辞”。他写道：——

我非常高兴新的《图论杂志》诞生了——但我不禁感到遗憾，德奈什未能活着看到图论如今的繁荣，而他为此做出了如此巨大的贡献。我本人是在高中时对图论产生兴趣的，当时我看到德奈什发表在一本面向高中生的数学杂志上的一篇论文。我读大一时，T 蒂博尔、G Grünwald和我一起上了德奈什的图论课程。我们三人很快就开始独立地“证明与猜想”。奇怪的是，在那个黑暗的年代，图论和组合分析竟如此不受重视。我父母的一位朋友，一位统计学家，曾这样评价德奈什：“他在自己的艺术中很伟大，但他的艺术如此渺小。”十年后，伟大的英国拓扑学家J H C Whitehead评价一位图论学家时说：“他在拓扑学的贫民窟里工作。”1938年我刚到普林斯顿时，惊讶地发现许多拓扑学家看不起四色问题，认为它是个不重要的枝节问题。[今天我们有了]“组合爆炸”。……越来越多的数学家意识到，在这个领域中还有许多美丽而出人意料的定理和理论有待发现……

德奈什未能活着看到图论兴趣的这种爆发，而这种爆发至少部分是由他的贡献所激发的。要理解他的死，我们必须回顾一些匈牙利的历史。匈牙利在德国的压力下，于1940年与德国签署了一项条约。从1941年7月起，匈牙利将许多犹太人移交给德国军队。然而，德奈什虽然是犹太人，却从小接受基督教教育，因此并未受到这场针对匈牙利犹太人的特定迫害的威胁。事实上，德奈什曾努力帮助受迫害的数学家。1944年9月，苏联军队进入匈牙利，政府与苏联签署了停战协定。德国对此感到愤怒，于10月15日入侵匈牙利，并扶植了一个由匈牙利国家社会主义党成员组成的傀儡政府。这个政府立即对所有匈牙利犹太人采取行动，突然间，德奈什因基督教身份而获得的保护一夜之间消失了。一场针对布达佩斯犹太人的恐怖统治开始了，德奈什不愿遭受面临的恐怖，选择了结束自己的生命。

Dénes Kőnig was the son of the famous Hungarian mathematician Gyula (Julius) Kőnig and Eliz Oppenheim (1853-1916). There were two boys in the family; Dénes had a brother György. Dénes received an excellent education in Budapest, attending one of the best schools, a Gymnasium, in the city. This Gymnasium, known as 'Minta', became the model for a new style of Hungarian high school. Of course, having an eminent mathematician as a father, Dénes was taught a lot of mathematics by his father, but he also had two excellent mathematics teachers at the Gymnasium, namely Manó Beke (1862-1946), who had studied under Felix Klein at Göttingen in 1892-93, and Miklós Szijártó, who had been a pupil of Loránd Eötvös. Dénes Kőnig had a most remarkable high school career for he achieved things that are basically unheard of for a school pupil. He had his first paper Elementary discussion of two maximum-minimum problems (Hungarian) published in 1899 while he was still at school and not yet fifteen years old. This is quite remarkable but a few others have achieved a similar feat. Perhaps even more remarkable is the fact that while he was at school he wrote a 61-page book, Mathematical Recreations (Hungarian). This was published in 1902, a year after he graduated from the Budapest 'Minta' Gymnasium, and a second volume was published in 1905 while he was still a university student. The Preface to the 1902 volume was written by Manó Beke who stressed the importance of mathematical recreation problems in complementing the school curriculum. He also pointed out that Kőnig included not only elementary problems but also included more advanced material. Beke writes:-

It was my great joy that my young friend, who is one of my dearest students, undertook to edit a booklet entitled "Mathematical entertainments". He has somewhat of a talent for this work. He undertook also to arouse an interest in mathematical problems in the school as well as out of school and an interest of readers in the general public, letting them deal with many mathematical questions exceeding the subjects to be taught in high-school. There are already a great many books in such an orientation; publications from abroad have the most remarkable collections in this genre. The author chose his subjects from those collections, and readers can find the sources listed at the end of the booklet. Even if a collection in the Hungarian language exists, it is much more elementary than the work here, and only includes rather commonplace arithmetic and geometry puzzles. This booklet includes not only such things, but also something beyond the elementary problems. It includes not only popular puzzling arithmetic problems including riddles which usually can be solved with linear equations, but also more remarkable works which were printed in foreign countries as mentioned above, and which lead us into the wonderful world of the numbers. ...

Both Beke and Szijártó were strong advocates of reforming mathematical education and this book by Kőnig was clearly seen as part of that project. After taking his baccalaureate examinations in 1902, he won first place in the Loránd Eötvös high school mathematics competition.

Later in 1902, Kőnig entered the University of Budapest where he spent four semesters. He then moved to Germany where he studied at the University of Göttingen for five semesters, returning to Budapest at Christmas 1906. While in Göttingen, he attended Hermann Minkowski's lectures on topology (called Analysis Situs at this time) in session 1904-05. Back in Budapest, he was advised by József Kürschák. He obtained his doctorate in 1907 from the Technical University of Budapest for his thesis Elementary Discussion of Rotations and Finite Rotation Group of a Space of Many Dimensions (Hungarian) [2]:-

Although Kőnig's dissertation treated geometrical problems, his supervisor Kürschák was also interested in mathematical recreations and graph theory. Kürschák's works are cited in two places in Kőnig's books on mathematical recreations published in 1902 and 1905.

After the award of his doctorate, Kőnig joined the staff of the Technische Hochschule in Budapest, where his father was professor. His initial appointment was as a demonstrator but in 1908 he became an assistant to the professor of mathematics. He published two joint papers with Alfred Haar in 1911, the paper On simply ordered sets (Hungarian) and its German translation as Über einfach geordnete Mengen Ⓣ. By 1910 he was ranked as the first assistant and in the following year became a docent. This meant that he could present his own lecture courses and he delivered the course Analysis Situs. Over the following years when he worked as a docent, Kőnig gave courses on Analysis Situs, Nomography, Real Numbers, Set Theory, Real Numbers and Functions, and Graph Theory. Although he did not give a course entitled Graph Theory until session 1927-28, nevertheless he had included chapters on graph theory in his Analysis Situs courses from 1911 onwards. From 1913-14 until 1927, in addition to teaching mathematics students, Kőnig also taught mathematics to students of Architecture and Chemical Engineering. He remained at the Technische Hochschule in Budapest until his death, becoming a full professor in 1935.

In the years 1918 and 1920 Kőnig published important books. The first was The Elements of Analysis Situs which appeared in 1918 [6]:-

He treats the topology of orientable 2-manifolds in a way accessible to beginners. This book was the first on this topic not only in domestic, but in the international mathematical literature as well. Prior to it there were neither textbooks nor monographs in this area (the only available literature was on Riemann surfaces). .... Several Hungarian mathematicians learned the topology of surfaces from Kőnig's book. Out of consideration for the beginner, Kőnig uses an intuitive approach. The book proves the main classification theorem and describes the various normal forms. The presentation is very careful. He often finds new approaches to concepts and proofs; in particular, he gives a simplified version of the reduction to normal form. For several proofs he also filled the gaps that existed in the literature.

The second book, which appeared in 1920, was Mathematics. Lectures at the Polytechnical University of Budapest for Students in Architecture and Chemical Engineering.

At Göttingen, Kőnig had been influenced by Minkowski's lectures on the four colour problem in the 1904-05 session [6]:-

In these, in addition to the characterisation of the topological properties of two-dimensional surfaces and the generation of various normal types, Minkowski intended to present a proof, given by Wemicke, of the four-colour conjecture. Only during the presentation of preliminary steps of the proof did it turn out that the proof was erroneous, and so only the proof of the five-colour theorem was presented.

These lectures contributed to Kőnig's growing interest in graph theory, but there were other reasons for his interest in the topic. He had studied the work of David Hilbert and Paul Gordan on invariant theory and they had referred to Julius Petersen's paper Die Theorie der regulären Graphs Ⓣ (1879). Kőnig saw that the use of graph theory had greatly helped visualise problems and help to solve them. He decided that he would try to make graph theory into a respectable mathematical discipline. Up till this time, the use of graphs was seen as an amusing way to present ideas to children rather than as a tool in solving deep problems. His first publications on graph theory, where he began his mission, were two papers published in Hungarian in 1911, namely Graphs on Two-sided Surfaces and The Genus of Graphs. In 1914 he attended the Congrès de philosophie mathématique in Paris where he gave a lecture in which he presented a graph theory result known today as the 'Theorem of Kőnig'. It is the first result in a topic which today is called matching theory. Because of the outbreak of World War I, the proceedings of this conference, containing Kőnig's paper, were not published until 1923. However, he published the 'Theorem of Kőnig' in a Hungarian paper of 1916 and also in a German paper Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre Ⓣ in the same year. This work on the factorisation of bipartite graphs relates closely to the marriage problem of Philip Hall. Kőnig's use of graphs to give a simpler proof of a determinant result of Georg Frobenius seems to have led to some hostility between the two men. Frobenius's reducible determinant theorem was published in 1912 while Kőnig's simpler proof, in terms of the perfect matchings of a certain bipartite graph, appeared in 1915 in his paper Line systems and determinants (Hungarian).

Kőnig's father, Gyula Kőnig, died in 1913. He had spent the last eight years of his life working on set theory, in particular on the continuum hypothesis. When Gyula died, a book he had been writing on set theory was almost, but not quite, complete. Dénes Kőnig completed the book Neue Grundlagen der Logik, Arithmetik und Mengenlehre Ⓣ adding a note to the Foreword already written by his father, saying that his father:-

... had worked for eight years [on the book], until the last day of his life. Only a few pages more were lacking.

After thanking József Kürschák for his help in finalising his father's manuscript, Kőnig writes:-

I also address my sincere thanks to Professor F Hausdorff. By undertaking the painful and unrewarding work of proofreading with me the whole book, he has fulfilled one of the last wishes of my father.

It was not only Gyula Kőnig who was interested in set theory, however, for Dénes Kőnig also made some important contributions to the topic. His interest in the topic began around 1908 when he attempted to give proofs of two of Felix Bernstein's theorems on the equivalence of sets without assuming the well-ordering principle. However, his interest in set theory was not unrelated to his interest in graph theory for he often translated his set theory results into graph theoretical terms. His work on Felix Bernstein's theorems led him to give several different versions of what today is called 'Kőnig's infinity lemma', namely that:-

... if there is no finite upper bound to the length of paths in a finitary tree, then there is at least one infinite path in the tree.

Forms of this lemma appear in Kőnig's papers of 1926 and one of 1927, namely Über eine Schlussweise aus dem Endlichen ins Unendliche Ⓣ which is entirely devoted to the infinity lemma. Miriam Franchella writes [4]:-

... to Kőnig's great credit, he had been able to isolate the lemma and to understand its wide applicability.

Kőnig's book, Theorie der endlichen und unendlichen Graphen Ⓣ, was published in 1936, and was a major factor in the growth of interest in graph theory worldwide. It was eventually translated into English under the title Theory of finite and infinite graphs (translated by R McCoart), Birkhauser, 1990. Keith Lloyd reviews the English translation in [10]:-

Dénes Kőnig's 'Theorie der endlichen und unendlichen Graphen' Ⓣ was originally published in Leipzig in 1936 and it was the first textbook on graph theory. It was reprinted by Chelsea in 1950 and even though many texts on the subject have appeared since then, Kőnig's book remains one which is still frequently cited. It is surprising, therefore, that it has taken more than half a century for an English edition to appear. The bulk of the present book consists of a translation by Richard McCoart of the whole of the original German text. ... Many of the topics dealt with by Kőnig are to be found in almost any text on the subject, for example, Euler trails, Hamiltonian cycles, mazes, trees, directed graphs and factorisations, but unlike many later authors, Kőnig considers infinite as well as finite graphs. As well as the translation, the English edition contains a 43-page Commentary by W S Tutte and a Biographical sketch of Kőnig by Tibor Gallai. In the former, Tutte summarises, chapter by chapter, the material treated by Kőnig and also discusses later developments, so providing an historical perspective on Kőnig's work.

We give the following quote from the Introduction to the 1936 book:-

Perhaps even more than to the interaction between mankind and nature, graph theory is based on the interaction of human beings with each other.

In the Preface to his 1936 book, Kőnig discusses whether graph theory is a branch of topology or a branch of combinatorics. He argues that it is the latter:-

... mainly because we attribute to the elements of a graph - vertices and edges - no geometrical content at all: the vertices are arbitrary distinguishable elements, and an edge is nothing other than a unification of its two endpoints. This abstract point of view - which Sylvester emphasized in 1873 - will be strictly maintained in our representation, with the exception of some examples and applications.

As to Kőnig's personality, we quote from [6]:-

He was a cheerful, sparkling man. He loved company; he enjoyed telling anecdotes. With his sarcastic humour he could entertain his company superbly. He liked his colleagues and was an indispensable participant in the coffee-house meetings of mathematicians.

He will always be thought of as one of the main originators of graph theory. He often began a lecture with the words:-

Graph theory is one of the most interesting of mathematical disciplines.

Paul Erdős wrote a 'Welcome' in the first issue of the Journal of Graph Theory in 1977. He wrote:-

I am very glad that the new 'Journal of Graph Theory' is born - but I cannot help feeling sorry that Dénes Kőnig did not live to see the present flowering of graph theory to which he contributed so much. I myself got interested in graph theory when I was in high school and saw a paper by Kőnig published in the mathematical magazine for high school students. When I was a freshman, T Gallai, G Grünwald and I took Kőnig's course on graph theory. All three of us very rapidly began to "prove and conjecture" independently. It is curious how little graph theory and combinatorial analysis was appreciated in those dark days. A friend of my parents, a statistician, once said about Dénes Kőnig: "He is great in his art but his art is so small". Ten years later J H C Whitehead, the great English topologist, said about a graph theorist: "He works in the slums of topology". When I first got to Princeton in 1938, I was surprised how many of the topologists looked down upon the four colour problem and considered it an unimportant side issue. [Today we have a] "combinatorial explosion". ... more and more mathematicians realise that there are very many beautiful and unexpected theorems and theories to be discovered in this field ...

Sadly Kőnig did not live to see this explosion of interest in graph theory which, at least in part, was motivated by his contributions. To understand his death we must look at some history of Hungary. Hungary, under pressure from Germany, signed a pact with Germany in 1940. From July 1941, Hungary handed many Jews over to the German forces. Kőnig, however, although Jewish, was brought up a Christian and so was not in danger from this particular persecution of Hungarian Jews. In fact Kőnig worked to help persecuted mathematicians. In September 1944, Soviet troops crossed into Hungary and the government signed an armistice with the Soviet Union. Angered by this, Germany invaded Hungary on 15 October and installed their own puppet government composed of members of the Hungarian National Socialist Party. This government moved immediately against all Hungarian Jews, and suddenly Kőnig's protection through his Christianity vanished overnight. A reign of terror was launched against the Jews of Budapest and, rather than suffer the horrors that faced him, Kőnig took his own life.