Abstract
A vertex subset R of an oriented graph \(\overrightarrow{G}\) is a relative oriented clique if each pair of non-adjacent vertices of R is connected by a directed 2-path. The relative oriented clique number \(\omega _{ro}(\overrightarrow{G})\) of \(\overrightarrow{G}\) is the maximum value of |R| where R is a relative oriented clique of \(\overrightarrow{G}\). Given a family \(\mathcal {F}\) of oriented graphs, the relative oriented clique number is \(\omega _{ro}(\mathcal {F}) = \max \{\omega _{ro}(\overrightarrow{G})|\overrightarrow{G} \in \mathcal {F}\}\). For the family \(\mathcal {P}_4\) of oriented triangle-free planar graphs, it was conjectured that \(\omega _{ro}(\mathcal {P}_4)=10\). In this article, we prove the conjecture.
This work is partially supported by the IFCAM project “Applications of graph homomorphisms” (MA/IFCAM/18/39) and ANR project HOSIGRA (ANR-17-CE40-0022).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We use this notation frequently to denote \(\alpha , \bar{\alpha } \in \{+,-\}\) and \(\alpha \ne \bar{\alpha }\). Our notation is a set theoretic equation whose solutions are the values that \(\alpha , \bar{\alpha }\) may take.
References
Courcelle, B.: The monadic second order logic of graphs VI: on several representations of graphs by relational structures. Discrete Appl. Math. 54(2), 117–149 (1994)
Das, S., Prabhu, S., Sen, S.: A study on oriented relative clique number. Discrete Math. 341(7), 2049–2057 (2018)
Klostermeyer, W.F., MacGillivray, G.: Analogues of cliques for oriented coloring. Discussiones Mathematicae Graph Theory 24(3), 373–388 (2004)
Kostochka, A.V., Sopena, É., Zhu, X.: Acyclic and oriented chromatic numbers of graphs. J. Graph Theory 24(4), 331–340 (1997)
Marshall, T.H.: Homomorphism bounds for oriented planar graphs. J. Graph Theory 55, 175–190 (2007)
Nandy, A., Sen, S., Sopena, É.: Outerplanar and planar oriented cliques. J. Graph Theory 82(2), 165–193 (2016)
Raspaud, A., Sopena, É.: Good and semi-strong colorings of oriented planar graphs. Inf. Process. Lett. 51(4), 171–174 (1994)
Sen, S.: A contribution to the theory of graph homomorphisms and colorings. Ph.D. thesis, Bordeaux University, France (2014)
Sopena, É.: Homomorphisms and colourings of oriented graphs: an updated survey. Discrete Math. 339(7), 1993–2005 (2016)
Sopena, É.: The chromatic number of oriented graphs. J. Graph Theory 25, 191–205 (1997)
Sopena, É.: Oriented graph coloring. Discrete Math. 229(1–3), 359–369 (2001)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Das, S.S., Nandi, S., Sen, S. (2020). The Relative Oriented Clique Number of Triangle-Free Planar Graphs Is 10. In: Changat, M., Das, S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science(), vol 12016. Springer, Cham. https://doi.org/10.1007/978-3-030-39219-2_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-39219-2_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-39218-5
Online ISBN: 978-3-030-39219-2
eBook Packages: Computer ScienceComputer Science (R0)