Crossing-critical graphs of high vertex degrees

Bc. Michal Korbela

Master's thesis

Abstract:

Nedávny výsledok Bokala a kol. konečne zodpovedal otázku ohľadom maximálneho stupňa vrcholov v c-crossing-critical grafoch. Pre každú hodnotu c<=12 existuje konštanta Dc taká, že každý c-crossing-critical graf má stupeň vrcholu nanajvýš Dc a pre každé c>=13 poskytli induktívnu konštrukciu c-crossing-critical grafov s ľubovolne vysokými stupňami vrcholov. Kým induktívna časť konštrukcie je viac menej jednoduchá, celá závisí na fakte, že jeden konkrétny 17-vrcholový graf má crossing number 13, pre čo poskytli len dôkaz počítačom. V tejto práci my poskytneme relatívne krátky samostatný ručný dôkaz.

Abstract:

A recent result of Bokal et al. finally gave an answer to the question of maximum vertex degrees in c-crossing-critical graphs. For each c ≤ 12, there exists a constant D c such that every c-crossing-critical graph has vertex degrees at most D c and for each c ≥ 13, they provided an inductive construction of c-crossing-critical graphs with arbitrarily high vertex degrees. While the inductive part of the construction is rather easy, it all relies on the fact that a certain 17-vertex base graph has crossing number 13, for which they provided only a machine-readable computer proof. We now provide a relatively short self-contained handwritten i.e. computer-free proof.

Keywords

graph drawing, crossing numbers, crossing-critical

Language used: English

Date on which the thesis was submitted / produced: 18. 5. 2021

Thesis defence

• Date of defence: 21. 6. 2021
• Supervisor: prof. RNDr. Petr Hliněný, Ph.D.
• Reader: PhD Ander Lamaison Vidarte

Masaryk University

Faculty of Informatics

Master programme / field:
Informatics / Theoretical Informatics