When I began to read mathematics at Hamburg as an undergraduate, I was able to study happily for three years without noticing the presence of Rudolf Halin. These were turbulent times, and Halin was keeping his head down. It troubled him deeply that students at that time seemed, occasionally, to be looking for a fight: while their plight always moved him, he found no way to communicate this in ways that the noisier ones amongst us, too, would hear. So few of us got to know him; and I was not one of these lucky ones.

After a few years at Hamburg, I went to Cambridge, initially for a year. My aim was to study graph theory, at last. When I sat in what must have been Andrew Thomason’s first graduate course, I was not a little surprised to be literally bombarded with two names: Halin and Mader. Andrew was teaching from Béla Bollobás’s Extremal Graph Theory, and giving us an animated account of these two young researchers’ ping-pong of results from the 1960s, each improving on the previous, which started with Halin’s seminal theorem that minimally k-connected graphs have a vertex of degree k [25].

Although Halin had early made a name for himself in structural finite graph theory, his heart was with the infinite. One of Wagner’s top students, he started his mathematical life by expanding the tool of ‘simplicial decompositions’ of graphs, which Wagner had developed to prove his now famous Equivalence Theorem (that Hadwiger’s conjecture for 5 is equivalent to the then 4-colour-conjecture), into a powerful theory for infinite graphs; see, e.g., [6,15,17,42,46,49]. Halin then applied this to obtain the first excluded minor theorems in infinite graph theory, to study Hadwiger’s conjecture for infinite cardinals, and so on: results several of which have remained unsurpassed to this day.

Another of Halin’s early papers that is still influential today is his 1973 study of the Automorphisms and endomorphisms of locally finite graphs [32]. When such a graph is connected, then every automorphism fixes either a finite subgraph or an end: a point at infinity in its Freudenthal compactification. This is a fundamental fact, for example, in the study of finitely generated groups via their Cayley graphs, on which the group elements themselves act as automorphisms.

Halin’s approach to ends differed radically from how they were previously conceived. I do not know whether Halin was aware of Freudenthal’s earlier work on the ends of groups, and later of more general locally compact spaces; he was certainly aware of Carathéodory’s Primenden. But while, for Carathéodory and Freudenthal, ends were something quite abstract, like boundary points in an inverse limit of compact spaces obtained by contracting the components of what is left after deleting a compact subspace, Halin exploited the discrete nature of graphs to define ends constructively: as equivalence classes of infinite paths that cannot be finitely separated [5].

Representing ends by concrete paths in this way has made them much more accessible to combinatorial analysis, a fact he used to great avail in his work, for example, on Menger-type connectivity between ends [33].

Halin worked in graph theory right from the start, an area which was just emerging at the time and barely recognised as mathematics. Yet he was already, and remained throughout his life, an archetypical mathematician. What spurred him on was the search for the essential. For example, he suggested [54] that the ‘essence’ of an infinite graph might be captured by declaring two graphs as equivalent if one could be obtained from the other in finitely many steps in such a way that of any two graphs adjacent in this finite sequence one was a contraction minor of the other, with finite branch sets. Are there ‘typical’ representatives of these equivalence classes that are easy to describe? Can we always make do with a sequence of length 2? Neither of these questions, to the best of my knowledge, has been answered.

Another example has been taken up more readily. Wagner’s simplicial decompositions of a graph, obtained recursively by splitting it along complete separators, will always lead to parts that fit together in a tree-like way that describes a coarse overall structure of the original graph. The drawback is that few graphs have enough complete separators to make this worthwhile. So why insist? Separations along complete separators are nested, and this is what produces the tree-structure. In modern terminology, every collection of nested separations of a graph G defines a tree-decomposition of G, and these are eminently useful even when the separators are not complete. Halin recognised this, and proved the first known theorems about tree-width and graph minors [38] nearly ten years before Robertson and Seymour reinvented these notions, gave them their modern name, and made them famous.

As a final example for how his search for the essential inspired Halin’s work let me mention his famous ‘end-faithful spanning tree’ problem. The simplest graphs that can have ends are trees, and so Halin asked whether every connected graph G had a spanning tree T whose end structure captured that of G in the sense that the natural map from the ends of T to those of G (inclusion, if we think of ends as sets of infinite paths) was bijective [5]. He proved this for countable G, and later for graphs without a topological \(K_{\aleph _0}\) minor [42]. The general problem remained open for over 25 years until Thomassen, and independently Seymour and Thomas, constructed counterexamples in 1989.

However, even this story has a happy ending. Another 25 years later it was shown that arbitrary connected graphs have end-faithful spanning trees after all—as long as we seek to represent not all their ‘Halin ends’ but just those that are ends also in the sense of Freudenthal. Fittingly, this deep result was obtained in a Hamburg Ph.D. thesis (2015), by one of Halin’s many mathematical grandchildren: Johannes Carmesin.

Rudolf Halin himself had twelve Ph.D. students, and he inspired many more. His search for the essential lives on. This volume, with contributions by many who admired Halin both for his work and as a person, bears witness to his legacy.

Reinhard Diestel, September 2016

1 Publications of Rudolf Halin

  1. [1]

    Wagner, K., Halin, R.: Homomorphiebasen von Graphenmengen. Math. Ann 147, 126–142 (1962)

  2. [2]

    Halin, R.,Jung, H.A.: Über Minimalstrukturen von Graphen, insbesondere von n-fach zusammenhängenden Graphen. Math. Ann. 152, 75–94 (1963)

  3. [3]

    Halin, R.: Über trennende Eckenmengen in Graphen und den Mengerschen Satz. Math. Ann. 157, 34–41 (1964)

  4. [4]

    Halin, R., Jung, H.A.: Charakterisierung der Komplexe der Ebene und der 2-Sphäre. Arch. Math. 15, 466–469 (1964)

  5. [5]

    Halin, R.: Über unendliche Wege in Graphen, Math. Ann. 157, 125–137 (1964)

  6. [6]

    Halin, R.: Über simpliziale Zerfällungen beliebiger (endlicher oder unendlicher) Graphen. Math. Ann. 156, 216–225 (1964)

  7. [7]

    Halin, R.: Über einen Satz von K. Wagner zum Vierfarbenproblem. Math. Ann. 153, 47–62 (1964)

  8. [8]

    Halin, R.: Bemerkungen über ebene Graphen. Math. Ann. 153, 38–46 (1964)

  9. [9]

    Halin, R.: Einige Bemerkungen über unendliche Graphen, Math. Nachr. 28, 365–385 (1964/1965)

  10. [10]

    Halin, R.: Über die Maximalzahl fremder unendlicher Wege in Graphen. Math. Nachr. 30, 63–85 (1965)

  11. [11]

    Halin, R.: Charakterisierung der Graphen ohne unendliche Wege. Arch. Math. 16, 227–231 (1965)

  12. [12]

    Halin, R.: Zu einem Problem von B. Grünbaum. Arch. Math. (Basel) 17, 566–568 (1966)

  13. [13]

    Halin, R.: Zur häufungspunktfreien Darstellung abzählbarer Graphen in der Ebene. Arch. Math. (Basel) 17, 239–243 (1966)

  14. [14]

    Halin, R.: Graphen ohne unendliche Wege. Math. Nachr. 31, 111–123 (1966)

  15. [15]

    Halin, R.: Unterteilungen vollständiger Graphen in Graphen mit unendlicher chromatischer Zahl. Abh. Math. Sem. Univ. Hambg. 31, 156–165 (1967)

  16. [16]

    Halin, R.: Zur Klassifikation der endlichen Graphen nach H. Hadwiger und K. Wagner. Math. Ann. 172, 46–78 (1967)

  17. [17]

    Halin, R.: Ein Zerlegungssatz für unendliche Graphen und seine Anwendung auf Homomorphiebasen. Math. Nachr. 33, 91–105 (1967)

  18. [18]

    Halin, R., Jung, H.A.: Note on isomorphisms of graphs. J. Lond. Math. Soc. 42, 254–256 (1967)

  19. [19]

    Halin, R.: Zum Mengerschen Graphensatz, Beiträge zur Graphentheorie (Kolloquium, Manebach, 1967), pp. 41–48. Teubner, Leipzig (1968)

  20. [20]

    Halin, R.: On the classification of finite graphs according to H. Hadwiger and K. Wagner. Theory of Graphs (Proc. Colloq., Tihany, 1966), pp. 161–167. Academic Press, New York (1968)

  21. [21]

    Halin, R.: Untersuchungen über minimale n-fach zusammenhängende Graphen. Math. Ann. 182, 175–188 (1969)

  22. [22]

    Halin, R.: Kreise beschränkter Länge in gewissen minimalen n-fach zusammenhängenden Graphen. Math. Ann. 183, 323–327 (1969)

  23. [23]

    Halin, R.: Zur Theorie der n-fach zusammenhägenden Graphen. Abh. Math. Sem. Univ. Hambg. 33, 133–164 (1969)

  24. [24]

    Halin, R.: On the structure of n-connected graphs. In: Recent Progress in Combinatorics (Proc. Third Waterloo Conf. on Combinatorics, 1968), pp. 91–102. Academic Press, New York (1969)

  25. [25]

    Halin, R.: A theorem on n-connected graphs. J. Combin. Theory 7, 150–154 (1969)

  26. [26]

    Halin, R.: Ecken n-ten Grades in minimalen n-fach zusammenhängenden Graphen. Abh. Math. Sem. Univ. Hamburg 35, 39–53 (1970)

  27. [27]

    Halin, R.: Die Maximalzahl fremder zweiseitig undendlicher Wege in Graphen. Math. Nachr. 44, 119–127 (1970)

  28. [28]

    Halin, R.: Über eine Klasse von Zerlegungen endlicher Graphen, Elektron. Informationsverarbeit. Kybernetik 6, 14–28 (1970)

  29. [29]

    Halin, R.: A colour problem for infinite graphs, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), pp. 123–127. Gordon and Breach, New York (1970)

  30. [30]

    Halin, R.: Unendliche minimale n-fach zusammenhängende Graphen. Abh. Math. Sem. Univ. Hambg. 36, 75–88 (1971) (Collection of articles dedicated to Lothar Collatz on his sixtieth birthday)

  31. [31]

    Halin, R.: Studies on minimally n-connected graphs, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), pp. 129–136. Academic Press, London (1971)

  32. [32]

    Halin, R.: Automorphisms and endomorphisms of infinite locally finite graphs. Abh. Math. Sem. Univ. Hambg. 39, 251–283 (1973)

  33. [33]

    Halin, R.: A note on Menger’s theorem for infinite locally finite graphs. Abh. Math. Sem. Univ. Hambg. 40, 111–114 (1974)

  34. [34]

    Halin, R.: Some path problems in graph theory. Abh. Math. Sem. Univ. Hambg. 44, 175–186 (1975) [(1976)]

  35. [35]

    Halin, R.: A problem in infinite graph-theory. Abh. Math. Sem. Univ. Hambg. 43, 79–84 (1975)

  36. [36]

    Halin, R.: A problem concerning infinite graphs, Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), pp. 233–237. Academia, Prague (1975)

  37. [37]

    Halin, R.: On chainlike decompositions of graphs. In: Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), p. 268, no. XV. Utilitas Math.,Winnipeg, Man., Congressus Numerantium (1976)

  38. [38]

    Halin, R.: S-functions for graphs. J. Geometry 8(1-2), 171–186 (1976)

  39. [39]

    Halin, R.: S-Funktionen für Graphen, Contributions to graph theory and its applications (Internat. Colloq., Oberhof, 1977), pp. 97–104. Tech. Hochschule Ilmenau, Ilmenau (1977)

  40. [40]

    Halin, R.: Abzählbare ebene Graphen, Beiträge zur geometrischen Algebra (Proc. Sympos., Duisburg, 1976), Lehrbücher Monograph. Geb. Exakten Wissensch., Math. Reihe, vol. 21, pp. 133-139. Birkhäuser, Basel, Boston, MA (1977)

  41. [41]

    Halin, R.: Systeme disjunkter unendlicher Wege in Graphen, Numerische Methoden bei Optimierungsaufgaben, Band 3 (Tagung, Math. Forschungsinst., Oberwolfach, 1976), Internat. Ser. Numer. Math., vol. 36, pp. 55–67. Birkhäuser, Basel (1977)

  42. [42]

    Halin, R.: Simplicial decompositions of infinite graphs. Advances in graph theory (Cambridge Combinatorial Conf., Trinity Coll., Cambridge, 1977). Ann. Discrete Math. vol. 3, pp. 93–109 (1978)

  43. [43]

    Halin, R.: Some topics concerning infinite graphs, Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Colloq. Internat., vol. 260, pp. 211–213. CNRS, Paris (1978)

  44. [44]

    Halin, R.: Über unendliche Graphen der Ebene, Numerische Methoden bei graphentheoretischen und kombinatorischen Problemen, Band 2 (Tagung, Math. Forschungsinst., Oberwolfach, 1978). Internat. Ser. Numer. Math., vol. 46, pp. 122–127. Birkhäuser, Basel, Boston (1979)

  45. [45]

    Halin, R.: A note on infinite triangulated graphs. Discrete Math. 31(3), 325–326 (1980)

  46. [46]

    Halin, R.: Simplicial decompositions: some new aspects and applications. Graph theory, (Cambridge 1981). North-Holland Math. Stud., vol. 62, pp. 101–110. North-Holland, Amsterdam, New York (1982)

  47. [47]

    Halin, R.: Some remarks on interval graphs. Combinatorica 2(3), 297–304 (1982)

  48. [48]

    Halin, R.: On the notion of infinite Hamiltonian graph. J. Graph Theory 7(4), 437–440 (1983)

  49. [49]

    Halin, R.: Simplicial decompositions and triangulated graphs. Graph theory and combinatorics (Cambridge 1983), pp. 191–196. Academic Press. London (1984)

  50. [50]

    Halin, R.: On the representation of triangulated graphs in trees. Eur. J. Combin. 5(1), 23–28 (1984)

  51. [51]

    Halin, R.: Universelle, Elemente, in Graphenklassen, die durch verbotene Konfigurationen definiert sind. Graphs, hypergraphs and applications (Eyba 1984), Teubner-Texte Math., vol. 73, pp. 48–51. Teubner, Leipzig (1985)

  52. [52]

    Diestel, R., Halin, R., Vogler, W.: Some remarks on universal graphs. Combinatorica 5(4), 283–293 (1985)

  53. [53]

    Halin, R.: Unendliche planare Graphen. Mitt. Math. Ges. Hambg. 11(6), 643–658 (1989)

  54. [54]

    Halin, R., Some problems and results on infinite graphs, Graph theory in memory of G. A. Dirac (Sandbjerg, 1985), Ann. Discrete Math., vol. 41, pp. 195–210. North-Holland, Amsterdam (1989)

  55. [55]

    Halin, R.: Das mathematische Werk K. Wagners, contemporary methods in graph theory. In: Bodendiek, R. (ed.) Honour of Prof. Dr. Klaus Wagner, pp. xi–xxii. Bibliographisches Inst., Mannheim (1990a)

  56. [56]

    Halin, R.: On infinite elements of homomorphism bases and a related class of graphs, Contemporary methods in graph theory. In: Bodendiek, R. (ed.) Honour of Prof. Dr. Klaus Wagner, pp. 323–352. Bibliographisches Inst., Mannheim (1990b)

  57. [57]

    Halin, R.: Fixed configurations in graphs with small number of disjoint rays, contemporary methods in graph theory. In: Bodendiek, R. (ed.) Honour of Prof. Dr. Klaus Wagner, pp. 639–649. Bibliographisches Inst., Mannheim (1990c)

  58. [58]

    Halin, R.: Bounded graphs, directions in infinite graph theory and combinatorics (Cambridge, 1989). Discrete Math. 95(1–3), 91–99 (1991)

  59. [59]

    Halin, R.: Tree-partitions of infinite graphs. Discrete Math. 97(1–3), 203–217 (1991)

  60. [60]

    Halin, R.: Lattices of cuts in graphs. Abh. Math. Sem. Univ. Hambg. 61, 217–230 (1991)

  61. [61]

    Halin, R.: Some finiteness results concerning separation in graphs, Special volume to mark the centennial of Julius Petersen’s “Die Theorie der regulären Graphs”. Part II. Discrete Math. 101(1–3), 97–106 (1992)

  62. [62]

    Halin, R.: Lattices related to separation in graphs, finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 411, pp. 153–167. Kluwer Acad. Publ., Dordrecht (1993)

  63. [63]

    Halin, R.: Minimization problems for infinite n-connected graphs. Combin. Probab. Comput. 2(4), 417–436 (1993)

  64. [64]

    Halin, R.: A note on partial 3-trees and homomorphism bases of graphs. Abh. Math. Sem. Univ. Hambg. 63, 29–36 (1993)

  65. [65]

    Halin, R.: Minimization problems for infinite n-connected graphs, Combinatorics, geometry and probability. A tribute to Paul Erdős. In: Bollobás, B., Thomason, A. (eds.) Proceedings of the Conference in Honor of Erdős’ 80th Birthday, (Cambridge, March 1993). Partial reprinting of Combinatorics, Probability and Computing, vol. 2 and 3, pp. 355-374. Cambridge University Press, Cambridge (1997)

  66. [66]

    Halin, R.: The structure of rayless graphs. Abh. Math. Sem. Univ. Hambg. 68, 225–253 (1998)

  67. [67]

    Halin, R.: A note on k-connected rayless graphs. J. Combin. Theory Ser. B 72(2), 257–260 (1998)

  68. [68]

    Halin, R.: A note on graphs with countable automorphism group. Abh. Math. Sem. Univ. Hambg. 70, 259–264 (2000a)

  69. [69]

    Halin, R.: Miscellaneous problems on infinite graphs. J. Graph Theory 35(2), 128–151 (2000b)

  70. [70]

    Halin, R.: On the cycle space of an infinite 3-connected graph. Results Math. 41(1–2), 106–113 (2002)

2 Books of Rudolf Halin

  1. [B1]

    Halin, R.: Graphentheorie. I, Erträge der Forschung (Research Results), vol. 138. Wissenschaftliche Buchgesellschaft, Darmstadt (1980)

  2. [B2]

    Halin, R.: Graphentheorie. II, Erträge der Forschung (Research Results), vol. 161. Wissenschaftliche Buchgesellschaft, Darmstadt (1981)

  3. [B3]

    Halin, R.: Graphentheorie, 2nd edn. Wissenschaftliche Buchgesellschaft, Darmstadt (1989)