Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees

Petr Hliněný, Michal Korbela

Abstract

A recent result of Bokal et al. [Combinatorica, 2022] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is c=13. The key to that result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high 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 the crossing number 13, which was originally verified only by a machine-readable computer proof. We provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to a remaining open question of this research area; we prove that for every c>=13 and integers d,q, there exists a c-crossing-critical graph with more than q vertices of each of the degrees 3,4,...,d.

On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees

Abstract

A recent result of Bokal et al. [Combinatorica, 2022] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is c=13. The key to that result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high 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 the crossing number 13, which was originally verified only by a machine-readable computer proof. We provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to a remaining open question of this research area; we prove that for every c>=13 and integers d,q, there exists a c-crossing-critical graph with more than q vertices of each of the degrees 3,4,...,d.

Paper Structure

This paper contains 8 sections, 14 theorems, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theorem \ref{['thm:main13alt']} without Assistance of a Computer
  3. Restricting optimal drawings of $G_{13}$
  4. Counting selected crossings in a drawing of $G_{13}$
  5. The Improved Construction for Theorem \ref{['thm:main13impro']}
  6. A generalized $13$-crossing-critical family
  7. Getting all possible vertex degrees
  8. Conclusion

Key Result

theorem 1

a) For each $1\leq c\leq12$, there exists a constant $D_c$ such that every $c$-crossing-critical graph has vertex degrees at most $D_c$. b) For each $c\geq13$ and every integers $q,d$, one can construct a $c$-crossing-critical graph which has at least $q$ vertices of degree at least $d$.

Figures (9)

  • Figure 1: An illustration of the two basic approaches to crossing-critical graph constructions. (a) The classical construction of $2$-crossing-critical graphs by Kochol in which the ends of the planar belt are joined twisted. (b) A construction of $c$-crossing-critical graphs ($c\geq3$, here $c=4$) by Hliněný in which the ends of the middle planar belt are joined straight, not twisted.
  • Figure 2: The inductive construction of $13$-crossing-critical graphs from Theorem \ref{['thm:main13']} (note that all $13$ depicted crossings are only between the blue edges). The edge labels in the picture represent the number of parallel edges between their end vertices (e.g., there are $7$ parallel edges between $x_1$ and $x_2$). Figure (a) defines the base graph $G_{13}$ of the construction, and (b) outlines the general construction which arbitrarily "duplicates" the two wedge-shaped gray subgraphs of $G_{13}$ and the gray vertices $x_1,x_2$, and possibly also "splits" the tips of the wedge-shaped gray subgraphs. See further Section \ref{['sec:improved']}.
  • Figure 3: Schematically, the two non-equivalent planar drawings of the subgraph $G_0$ (red and gray) of $G_{13}$. This is used in the proof of Lemma \ref{['lem:redgrayplanar']}.
  • Figure 4: The graph $G_{13}^{(1\!/2,\,1\!/2,\,1)}$.
  • Figure 5: Two schematic drawings of the graph $G$ from the proof of Theorem \ref{['thm:13altcrit']}. The dashed lines show alternative routings of some of the edges, and one may straightforwardly split the gray vertices $y^*$ and/or $y^i$ in order to obtain the full drawing of $G$ as required. a) A drawing with $13$ crossings which drop down to $12$ crossings after deleting any one of the edges $y^*x^i$ or $x^iy^i$. b) A drawing with $18$ crossings which drop down to $12$ crossings after deleting any one of the edges $w_1^iw_2^i$, $x^iw_1^i$, $w_4^iw_1^i$ or $w_2^iw_3^i$.
  • ...and 4 more figures

Theorems & Definitions (26)

  • definition 1
  • theorem 1: Bokal, Dvořák, Hliněný, Leaños, Mohar and Wiedera DBLP:journals/combinatorica/BokalDHLMW22
  • theorem 2: a computer-free alternative to Theorem \ref{['thm:main13']}(b)
  • theorem 3
  • proposition 1: folklore
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • ...and 16 more