Gert Sabidussi

Fixed elements of infinite trees

Abstract It is shown that any infinite tree not containing a ray has a fixed vertex or a fixed edge. The same also holds for trees with rays (not containing a subdivision of the dyadic tree) provided there are at least three ends of maximal order.

Julius Petersen 1839–1910 a biography

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. The person Julius Petersen Childhood and youth (1839-1871) Geometric constructions (1866-1879) The doctoral dissertation (1871) Social and economic engagement (1871-1877) Cryptography (1875) The theory of algebraic equations (1877) Docent at the Polytechnical School. Mechanics (1871-1887) Style of exposition and method of research Miscellaneous papers (1870-1890) Models and instruments (1887-1895) Invariant theory and graph theory (1888-1899) Professor at Copenhagen University (1887-1909) Inspector of the Learned Schools (1887-1900) Function theory, latin squares and number theory (1888-1909) Last years (1908-1910) References 9 12 14 18 23 28 29 34 37 38 47 49 67 69 75 78 79

Maximum independent sets in 3- and 4-regular Hamiltonian graphs

Smooth 4-regular Hamiltonian graphs are generalizations of cycle-plus-triangles graphs. While the latter have been shown to be 3-choosable, 3-colorability of the former is NP-complete. In this paper we first show that the independent set problem for 3-regular Hamiltonian planar graphs is NP-complete, and using this result we show that this problem is also NP-complete for smooth 4-regular Hamiltonian graphs. We also show that this problem remains NP-complete if we restrict the problem to the existence of large independent sets (i.e., independent sets whose size is at least one third of the order of the graphs).

Deeply asymmetric planar graphs

It is proved that by deleting at most 5 edges every planar (simple) graph of order at least 2 can be reduced to a graph having a non-trivial automorphism. Moreover, the bound of 5 edges cannot be lowered to 4.

Embedding graphs as isometric medians

We show that every connected graph can be isometrically (i.e., as a distance preserving subgraph) embedded in some connected graph as its median. As an auxiliary result we also show that every connected graph is an isometric subgraph of some Cayley graph.

Transforming Eulerian trails

Abstract In this paper a set of transformations (κ-transformations) between eulerian trails is investigated. It is known that two arbitrary eulerian trails can be transformed into each other by a sequence of κ-transformations. For compatible eulerian trails the set of κ-transformations is augmented by the set of κ-detachments and κ-absorptions. This augmented set is capable of transforming two arbitrary P -compatible eulerian trails ( P is an edge partition system) into each other. This result is applied to A -trails, alternating eulerian trails and digraphs.

Correspondence between Sylvester, Petersen, Hilbert and Klein on invariants and the factorisation of graphs 1889–1891

Abstract A collection of 47 letters shedding some light on the background and origin of Petersens famous paper on graph factorisation, and on his abortive collaboration with James Joseph Sylvester.

Factorizations of 4-Regular Graphs and Petersen's Theorem

On the basis of the observation that a 3-regular graph has a perfect matching if and only if its line graph has a triangle-free 2-factorisation, we show that a connected 4-regular graph has a triangle-free 2-factorisation, provided it has no more than two cut-vertices belonging to a triangle. This result is equivalent to Petersen?s theorem about the existence of perfect matchings in 3-regular graphs.

Binary invariants and orientations of graphs

It is well known that Julius Petersen’s famous paper on graph factorisation [9] has its origin in a problem which arose in connection with Hilbert’s proof of the Finite Basis Theorem for the invariants of binary forms [3]. Once one accepts the graph theoretic framework, the reformulation of the algebraic problem is straightforward and needs only a minimum of explanation-at least in the context of a hundred years ago-and this is precisely what Petersen provides in his paper. It is, however, unfortunate that he chose not to give any indication of the motivation for the fundamental idea of using graphs in connection with invariants, except in a very oblique way by acknowledging that Sylvester had also worked on the problem and that there had been some correspondence between them ([9, p. 1941, [13]). Indeed, Petersen’s whole approach is based on an observation made by Sylvester in 187~ignored by most invariant theorists at the time-to the effect that regular graphs contain essentially the same information as invariants of binary forms, in the sense that there is a natural map which assigns to each regular graph of order n an invariant of the binary form of order n, and that the invariants so obtained additively generate all invariants [14]. Instead of the unwieldy invariants one can therefore study the intuitively much more accessible graphs. In 1891 this was by no means self-evident, and Petersen’s silence on this point may well have contributed to the fact that his paper remained an isolated occurrence. In the present paper we shall take as our point of departure a problem which lies at the heart of the correspondence between graphs and invariants, and to which Petersen had given a certain amount of thought. The map graphs ---, invariants is many-to-one; it is therefore of interest to know which graphs lie in its ‘kernel’, i.e., give rise to the zero invariant. Petersen recognised the importance

Julius Petersen annotated bibliography

Abstract This is the first comprehensive bibliography of Julius Petersens papers and books, covering not only his mathematical works, but also his contributions to economics, social science, physics and education.