On 13-Crossing-Critical Graphs with Arbitrarily Large Degrees

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

Structure and Generation of Crossing-Critical Graphs

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For c=1 there are only two such graphs without degree-2 vertices, K_5 and K_{3,3}, but for any fixed c>1 there exist infinitely many c-crossing-critical graphs. It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every c-crossing-critical graph can be obtained from a c-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the c-crossing-critical graphs of at most given order n in polynomial time per each generated graph

Crossing-number critical graphs have bounded path-width

AbstractThe crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr(G−e)<cr(G) for all edges e of G. We prove that crossing-critical graphs have “bounded path-width” (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined in a linear way on small cut-sets. Equivalently, a crossing-critical graph cannot contain a subdivision of a “large” binary tree. This assertion was conjectured earlier by Salazar (J. Geelen, B. Richter, G. Salazar, Embedding grids on surfaces, manuscript, 2000)

Crossing-Critical Graphs and Path-Width

. The crossing number cr(G) of a graph G, is the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr(G e) &lt; cr(G) for all edges e of G. G. Salazar conjectured in 1999 that crossing-critical graphs have pathwidth bounded by a function of their crossing number, which roughly means that such graphs are made up of small pieces joined in a linear way on small cut-sets. That conjecture was recently proved by the author [9]. Our paper presents that result together with a brief sketch of proof ideas. The main focus of the paper is on presenting a new construction of crossing-critical graphs, which, in particular, gives a nontrivial lower bound on the path-width. Our construction may be interesting also to other areas concerned with the crossing number.