IWOCA 2018: Combinatorial Algorithms pp 273-284

The Crossing Number of Seq-Shellable Drawings of Complete Graphs

• Petra Mutzel
• Lutz Oettershagen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)

Abstract

The Harary-Hill conjecture states that for every $$n\ge 3$$ the number of crossings of a drawing of the complete graph $$K_n$$ is at least
\begin{aligned} H(n) := \frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \text {.} \end{aligned}
So far, the conjecture could only be verified for arbitrary drawings of $$K_n$$ with $$n\le 12$$. 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 $$K_{11}$$, therefore the class of seq-shellable drawings strictly contains the class of bishellable drawings.

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