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Rooted Structures in Graphs Technische Universität Ilmenau Institut für Mathematik Fachgebiet Diskrete Mathematik und Algebra Dissertation zur Erlangung des akademischen Grades Dr. rer. nat. Rooted Structures in Graphs A Project on Hadwiger’s Conjecture, Rooted Minors, and Tutte Cycles vorgelegt der Fakultät für Mathematik und Naturwissenschaften der Technischen Universität Ilmenau von Samuel Mohr 1. Gutachter: Prof. Dr. Matthias Kriesell 2. Gutachter: Prof. Dr. Henning Bruhn-Fujimoto 3. Gutachter: Dr. Nicolas Trotignon Tag der Einreichung: 3. April 2020 Tag der wissenschaftlichen Aussprache: 24. August 2020 urn:nbn:de:gbv:ilm1-2020000294 Dedication To T. Abstract Rooted Structures in Graphs Dissertation by Samuel Mohr Hugo Hadwiger conjectured in 1943 that each graph G with chromatic number χ(G) has a clique minor on χ(G) vertices. Hadwiger’s Conjecture is known to be true for graphs with χ(G) ≤ 6, with χ(G) = 5 and χ(G) = 6 being merely equivalent to the Four-Colour-Theorem. Even Paul Erdős (1980) stated that it is “one of the deepest unsolved problems in graph theory”, thus reinforcing the extreme nature and difficulty of this conjecture. One promising approach to tackle Hadwiger’s Conjecture is to bound the number of colourings of considered graphs and, in particular, to consider uniquely optimally colourable graphs. Matthias Kriesell started considerations on graphs with a Kempe colouring, that is graphs with a proper colouring such that the subgraph induced by the union of any two colour classes is connected. Evidently, the optimal colouring of a uniquely colourable graph is a Kempe colouring. A transversal of a graph’s Kempe colouring is a vertex set containing exactly one vertex from each colour class. For each transversal T of a Kempe colouring, there is a system of edge- disjoint paths between all vertices from T . The question arises, whether it is possible to show the existence of a clique minor of the same size as T such that each bag contains one vertex from T . We call a minor with the aforementioned property a rooted minor. An affirmative answer to this question, which was conjectured by Matthias Kriesell (2017), would imply Hadwiger’s Conjecture for uniquely colourable graphs. Beyond that, there are more reasons to study this conjecture of Kriesell. On one hand, there are graphs with a Kempe colouring using far more colours than their chromatic number. On the other hand, many proofs of partial results on Hadwiger’s Conjecture seem to leave no freedom for prescribing vertices in the clique minor at the expense of forcing any pair of colour classes to be connected. The question is known to be true for |T | ≤ 4 by a result of Fabila-Monroy and Wood. In this thesis, we confirm it for line graphs and for |T | = 5 if T induces a connected subgraph. For certain graph classes, it emerges that a relaxation of the problem holds. But in general, it turns out that this relaxation is false. Thus, it is not sufficient to force any two transversal vertices to be connected by a 2-coloured path in order to obtain a rooted minor. Based on rooted minors and results built up from connectivity conditions, the question arose whether it is possible to reduce graphs, in which a certain set of vertices is highly connected, to some basic structure representing the connectivity. This question is investigated in this Abstract ii thesis and answered with two best-possible theorems. The first one elucidates that if aset of vertices, let us call it X, cannot be separated with less than c vertices (c ≤ 3), then there exists a c-connected topological minor rooted at X. In case of c = 4, a second result ensures that it is at least possible to obtain a 4-connected rooted minor, whereas larger connectivities do not lead to highly connected minors in general. These theorems are applied in this thesis to achieve results about subgraphs containing the set X by deriving them from various results regarding spanning subgraphs. For instance, a theorem of Barnette, stating that a 3-connected planar graph contains a spanning tree of maximum degree at most 3, can easily translated; i.e. there exists a tree with maximum degree at most 3 containing X. Since we cannot guarantee 4-connected topological minors rooted at X, more effort is needed to establish cycles spanning X in planar graphs. To escape this barrier, the theory of Tutte paths is adapted. In 1956, William Tutte broke the long-standing open problem whether 4-connected planar graphs are hamiltonian and, thereby, extended a theorem of Whitney stating that 4-connected triangulations are hamiltonian. The underlying idea is to prove that planar graphs contain a cycle such that components outside the cycle have a limited number of attaching points to the cycle. In particular, this number of shared attachment vertices is bounded by 3, immediately implying Tutte’s result. By a construction of Moon and Moser it is known that there exist 3-connected planar graphs and a positive constant c such that the length of a longest cycle (circumference) is c · |V (G)|log3 2 of such a graph G. Several years later, this formula was confirmed to be a lower bound by Chen and Yu. We study a slight weakening of the 4-connectedness of planar graphs. A 3-connected graph is essentially 4-connected if all separators of order 3 only split single vertices from the remaining graph. If we consider essentially 4-connected planar graphs instead of 3-connected ones, then the minimum circumference in this graph class is described by a linear function in the number of vertices, as Jackson and Wormald showed first. In this thesis, we increase the previously best known coefficient of the linear bound, which was published by Fabrici, Harant, 2 and Jendroľ. Moreover, the best-possible bound of 3 (n + 4) for essentially 4-connected triangulations on n vertices is proved. Finally, the thesis explores the class of 1-planar graphs and studies their longest cycles. A graph is 1-planar if it has an embedding into the plane such that each edge is crossed at most once. This class, in contrast to planar graphs, differs with regard to many aspects. The decision problem of 1-planarity is NP-complete and there is more than one way to construe maximal 1-planar graphs. In this thesis, an analogue to Whitney’s theorem is proved; i.e., that 3-connected maximal 1-planar graphs have a sublinear circumference just as their planar relatives, and that 4-connected maximal 1-planar graphs are hamiltonian. In the non-maximal case, Tutte’s theorem cannot be translated directly; we will construct a family of 5-connected 1-planar graphs in the thesis that is far from being hamiltonian. Whether each 6-connected 1-planar graph is hamiltonian remains open. Some ideas on how an approach to this question using an extended Tutte theory could look like is finally touched upon in the outlook. Zusammenfassung Gewurzelte Strukturen in Graphen Dissertation von Samuel Mohr Der Schweizer Mathematiker Hugo Hadwiger stellte 1943 die Vermutung auf, dass jeder Graph G mit chromatischer Zahl χ(G) einen vollständigen Minoren auf χ(G) Knoten besitzt. Diese Vermutung ist für alle Graphen mit χ(G) ≤ 6 wahr, wobei für die Fälle χ(G) = 5 und χ(G) = 6 nur die Äquivalenz zum Vierfarbensatz bekannt ist. Paul Erdős (1980) nannte es eines der weitreichendsten ungelösten Probleme der Graphentheorie und bekräftigte die fehlende Greifbarkeit und Schwere dieser Vermutung. Ein vielversprechender Ansatz zu neuen Erkenntnissen über Hadwigers Vermutung ist eine Beschrän- kung der Anzahl der Färbungen. Dabei kann insbesondere die Untersuchung eindeutig färbbarer Gra- phen helfen. Um darüber Ergebnisse zu erzielen, startete Matthias Kriesell erste Untersuchungen von Graphen mit einer Kempe-Färbung. Das sind Graphen mit einer speziellen Färbung, bei denen die Vereinigung je zweier Farbklassen einen zusammenhängenden Untergraphen induziert. Offensichtlich ist die optimale Färbung eines eindeutigen Graphens auch eine Kempe-Färbung. Eine Transversale einer Kempe-Färbung eines Graphen ist eine Knotenmenge, die genau einen Knoten aus jeder Farbklasse enthält. Für jede Transversale T einer Kempe-Färbung gibt es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T . Es stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen Minoren derselben Größe wie T zu gewährleisten, sodass jede Tasche genau einen Knoten aus T enthält. Wir nennen einen Minor mit den oben beschriebenen Eigenschaften einen gewurzelten Minor. Eine positive Beantwortung dieser Frage, deren Bejahung bereits von Matthias Kriesell (2017) vermutet wurde, würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. Es gibt aber noch weitere Gründe, diese Vermutung zu untersuchen. Einerseits gibt es Graphen mit einer Kempe-Färbung, die wesentlich mehr Farben als die chromatische Zahl enthält. Andererseits lassen viele Beweise bekannter Teilresultate zu Hadwigers Vermutung keinen Spielraum, um Knoten in den Taschen der Minoren vorzuschreiben, auch nicht unter der Forderung, dass je zwei Farbklassen zusammenhängend sind. Die Frage lässt sich positiv für |T | ≤ 4 durch ein Resultat von Fabila-Monroy und Wood beantworten. In dieser Dissertation wird sie zudem für Kantengraphen und für |T | = 5 verifiziert, sofern T einen zusammenhängenden Teilgraphen induziert. Für bestimmte Gra- phenklassen wird sich zeigen, dass eine Verallgemeinerung des Problems gilt. Jedoch stellt sich heraus, dass die Gültigkeit der Verallgemeinerung im Allgemeinen nicht bestätigt werden kann. Somit ist es nicht ausreichend, zu fordern, dass je zwei Transversalknoten auf einem zweigefärbten Weg liegen, um einen gewurzelten Minor zu erhalten. Zusammenfassung iv Basierend auf gewurzelten Minoren und verschiedenen Sätzen, die auf Zusammenhangsbedin- gungen aufbauen, stellt sich die Frage, ob es möglich ist, Graphen, in denen eine bestimmte Menge an Knoten hoch zusammenhängend ist, auf gewisse einfachere Strukturen zu reduzie- ren, die lediglich den Zusammenhang repräsentieren.
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