Infinite families of crossing‐critical graphs with prescribed average degree and crossing number

Abstract

Širan constructed infinite families of ▫$k$▫-crossing-critical graphs for every ▫$k ge 3$▫ and Kochol constructed such families of simple graphs for every ▫$k ge 2$▫. Richter and Thomassen argued that, for any given ▫$k ge 1$▫ and ▫$r ge 6$▫, there are only finitely many simple ▫$k$▫-crossing-critical graphs with minimum degree ▫$r$▫. Salazar observed that the same argument implies such a conclusion for simple ▫$k$▫-crossing-critical graphs of prescribed average degree ▫$r > 6$▫. He established existence of infinite families of simple ▫$k$▫-crossing-critical graphs with any prescribed rational average degree ▫$r in [4,6)$▫ for infinitely many ▫$k$▫ and asked about their existence for ▫$r in (3,4)$▫. The question was partially settled by Pinontoan and Richter, who answered it positively for ▫$r in (3frac{1}{2},4)$▫. The present contribution uses two new constructions of crossing critical simple graphs along with the one developed by Pinontoan and Richter to unify these results and to answer Salazar\u27s question by the following statement: for every rational number ▫$r in (3,6)$▫ there exists an integer ▫$N_r$▫, such that, for any ▫$k > N_r$▫, there exists an infinite family of simple 3-connected crossing-critical graphs with average degree ▫$r$▫ and crossing number ▫$k$▫. Moreover, a universal lower bound on ▫$k$▫ applies for rational numbers in any closed interval ▫$I subset (3,6)$▫

On Degree Properties of Crossing-Critical Families of Graphs

“…They were used already in the early papers on infinite families of crossing-critical graphs by Kochol [13] and Richter and Thomassen [17], although they were formalized only in the work of Pinontoan and Richter [15,16], answering Salazar's question [18] on average degrees in infinite families of k-crossing-critical graphs. Bokal built upon these results to fully settle Salazar's question when combining tiles with zip product [3]. Also a recent result that all large 2-crossing-critical graphs are composed of large multi-sets of specific 42 tiles [5] demonstrates that tiles are intimately related to crossing-critical graphs.…”

“…Also a recent result that all large 2-crossing-critical graphs are composed of large multi-sets of specific 42 tiles [5] demonstrates that tiles are intimately related to crossing-critical graphs. In this section, we summarize the known results from [3,5,15], which we need for our constructions.…”

“…The gadgets we use are twisted pairs of paths, guaranteeing one crossing each, and staircase strips of width n, guaranteeing The disjointness of a twisted pair {P, Q} implies one crossing in any tile drawing of T . This is generalized to twisted families in the following lemma: Lemma 2.9 ( [3]). Let W be a twisted family in a tile T , such that no edge occurs in two distinct paths of ∪W.…”

“…Among the most studied such properties are those related to vertex degrees in the critical families, see [3,6,8,11,18]. Often the research focused on the average degree a k-crossing-critical family may havethis rational number clearly falls into the interval [3,6] if we forbid degree-2 vertices. It is now known [8] that the true values fall into the open interval (3,6), and all the rational values in this interval can be achieved [3].…”

Characterizing all graphs with 2-exceptional edges

“…The approach can be abstracted into the following steps, which allow to decompose an abstract graph with properties of interest into smaller pieces called tiles. The tiles are a tool often applied in the investigation of crossing critical graphs [2,3,4,7,9,10,11]. The method structures arguments as follows:…”