Bishellable drawings of $K_n$

B.M. Ábrego, O. Aichholzer, S. Fernández-Merchant, D. McQuillan, B. Mohar, P. Mutzel, P. Ramos, R.B. Richter, B. Vogtenhuber

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Abstract

In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b leq s-2$, also $b$-bishellable. Our main result is that $(lfloor n2 rfloor-2)$-bishellability also guarantees, with a simpler proof than for $s$-shellability, that a drawing has at least $H(n)$ crossings. We exhibit a drawing of $K_11$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved.
Original languageEnglish
Title of host publicationProc. XVII Encuentros de Geometría Computacional
Place of PublicationAlicante, Spain
Pages17-20
Number of pages4
Publication statusPublished - 2017

Cite this

Ábrego, B. M., Aichholzer, O., Fernández-Merchant, S., McQuillan, D., Mohar, B., Mutzel, P., ... Vogtenhuber, B. (2017). Bishellable drawings of $K_n$. In Proc. XVII Encuentros de Geometría Computacional (pp. 17-20). Alicante, Spain.

Bishellable drawings of $K_n$. / Ábrego, B.M.; Aichholzer, O.; Fernández-Merchant, S.; McQuillan, D.; Mohar, B.; Mutzel, P.; Ramos, P.; Richter, R.B.; Vogtenhuber, B.

Proc. XVII Encuentros de Geometría Computacional. Alicante, Spain, 2017. p. 17-20.

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Ábrego, BM, Aichholzer, O, Fernández-Merchant, S, McQuillan, D, Mohar, B, Mutzel, P, Ramos, P, Richter, RB & Vogtenhuber, B 2017, Bishellable drawings of $K_n$. in Proc. XVII Encuentros de Geometría Computacional. Alicante, Spain, pp. 17-20.
Ábrego BM, Aichholzer O, Fernández-Merchant S, McQuillan D, Mohar B, Mutzel P et al. Bishellable drawings of $K_n$. In Proc. XVII Encuentros de Geometría Computacional. Alicante, Spain. 2017. p. 17-20
Ábrego, B.M. ; Aichholzer, O. ; Fernández-Merchant, S. ; McQuillan, D. ; Mohar, B. ; Mutzel, P. ; Ramos, P. ; Richter, R.B. ; Vogtenhuber, B. / Bishellable drawings of $K_n$. Proc. XVII Encuentros de Geometría Computacional. Alicante, Spain, 2017. pp. 17-20
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T1 - Bishellable drawings of $K_n$

AU - Ábrego, B.M.

AU - Aichholzer, O.

AU - Fernández-Merchant, S.

AU - McQuillan, D.

AU - Mohar, B.

AU - Mutzel, P.

AU - Ramos, P.

AU - Richter, R.B.

AU - Vogtenhuber, B.

PY - 2017

Y1 - 2017

N2 - In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b leq s-2$, also $b$-bishellable. Our main result is that $(lfloor n2 rfloor-2)$-bishellability also guarantees, with a simpler proof than for $s$-shellability, that a drawing has at least $H(n)$ crossings. We exhibit a drawing of $K_11$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved.

AB - In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b leq s-2$, also $b$-bishellable. Our main result is that $(lfloor n2 rfloor-2)$-bishellability also guarantees, with a simpler proof than for $s$-shellability, that a drawing has at least $H(n)$ crossings. We exhibit a drawing of $K_11$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved.

M3 - Conference contribution

SP - 17

EP - 20

BT - Proc. XVII Encuentros de Geometría Computacional

CY - Alicante, Spain

ER -