On Degree Properties of Crossing-Critical Families of Graphs

Abstract

Answering an open question from 2007, we construct infinite k-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large k. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers D such that min(D) ≥ 3 and 3, 4 ∈ D, and for any sufficiently large k, there exists a k-crossing-critical family such that the numbers in D are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both D and some average degree in the interval (3, 6) are prescribed, k-crossing-critical families exist for any sufficiently large k.

Counting Hamiltonian Cycles in 2-Tiled Graphs

“…Salazar's question was resolved by Bokal, again using tiles. They were instrumental in further studies on degrees in crossing critical graphs by Hliněný [34], whose most complete results so far are published in [35]. Most of this desire for understanding the degrees in crossing-critical graphs was inspired by a conjecture of Richter that c-crossing-critical graphs have their maximum degree bounded from above by a function of c. The conjecture was first existentially disproved by Dvořák and Mohar [36].…”

“…In this section, we introduce the concept of a tile that was introduced in [32] and later redesigned in [42] and applied in [8,35] and the k-tiled graphs. We use the notation from [8].…”

Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

“…The crossing number ( ) of a graph is the minimum number of crossing points of edges in a drawing of in the plane. For ∈ ℕ, a graph is called -crossing-critical, if ( ) ≥ but, for every edge of , ( − ) < [1]. In general, for a graph , the minimum number of pairwise crossings of edges among all drawings of in the plane is the crossing number of and is denoted by ( ).…”

“…These reasons can be understood via so called -crossing-critical graphs. For ∈ ℕ, a graph is called -crossing-critical, if ( ) ≥ , but for every edge of , ( − ) < [1].…”

“…According to [1], a principal tool used in construction of crossing-critical graphs are tiles. Tiles are essentially graphs equipped with two sequences of vertices that are identified among tiles or within a tile in order to, respectively, form new tiles or tiled graphs.…”