The crossing number of seq-shellable drawings of complete graphs

Abstract

The Harary-Hill conjecture states that for every n≥3 the number of crossings of a drawing of the complete graph Kn is at least (Formula Presented) So far, the conjecture could only be verified for arbitrary drawings of Kn with n≤12. In recent years, progress has been made in verifying the conjecture for certain classes of drawings, for example 2-page-book, x-monotone, x-bounded, shellable and bishellable drawings. Up to now, the class of bishellable drawings was the broadest class for which the Harary-Hill conjecture has been verified, as it contains all beforehand mentioned classes. In this work, we introduce the class of seq-shellable drawings and verify the Harary-Hill conjecture for this new class. We show that bishellability implies seq-shellability and exhibit a non-bishellable but seq-shellable drawing of K11, therefore the class of seq-shellable drawings strictly contains the class of bishellable drawings.