Crossing-critical graphs with large maximum degree

A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number k is that they have bounded bandwidth. A weaker well-known conjecture is that their maximum degree is bounded in terms of k. In this note we disprove these conjectures for every k ≥ 171, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree. A graph is k-crossing-critical (or simply k-critical) if its crossing number is at least k, but every proper subgraph has crossing number smaller than k. Using the Excluded Grid Theorem of Robertson and Seymour [8], it is not hard to argue that k-crossing-critical graphs have bounded tree-width [2]. However, all known constructions of crossing-critical graphs suggested that their structure is “path-like”. Salazar and Thomas conjectured (cf. [2]) that they have bounded path-width. This problem was solved by Hliněný [3], who proved that the path-width of k-critical graphs is bounded above by 2f(k), where f(k) = (432 log2 k + 1488)k3 + 1. In the late 1990’s, two other conjectures were proposed (see [7] or [6]). ∗Supported in part through a postdoctoral position at Simon Fraser University. †On leave from: Institute of Theoretical Informatics, Charles University, Prague, Czech Republic. ‡Supported in part by the Research Grant P1–0297 of ARRS (Slovenia), by an NSERC Discovery Grant (Canada) and by the Canada Research Chair program. §On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.