A class of graphs with large rankwidth

Abstract

We describe several graphs with arbitrarily large rankwidth (or equivalently with arbitrarily large cliquewidth). Korpelainen, Lozin, and Mayhill [Split permutation graphs, Graphs and Combinatorics, 30(3):633-646, 2014] proved that there exist split graphs with Dilworth number 2 with arbitrarily large rankwidth, but without explicitly constructing them. We provide an explicit construction. Maffray, Penev, and Vu\v{s}kovi\'c [Coloring rings, Journal of Graph Theory 96(4):642-683, 2021] proved that graphs that they call rings on $n$ sets can be colored in polynomial time. We show that for every fixed integer $n\geq 3$, there exist rings on $n$ sets with arbitrarily large rankwidth. When $n\geq 5$ and $n$ is odd, this provides a new construction of even-hole-free graphs with arbitrarily large rankwidth.