Abstract
A simple signed graph \((G, \varSigma )\) is a simple graph with a \(+\)ve or a −ve sign assigned to each of its edges where \(\varSigma \) denotes the set of −ve edges. A cycle is unbalanced if it has an odd number of −ve edges. A vertex subset R of \((G, \varSigma )\) is a relative signed clique if each pair of non-adjacent vertices of R is part of an unbalanced 4-cycle. The relative signed clique number \(\omega _{rs}((G, \varSigma ))\) of \((G,\varSigma )\) is the maximum value of |R| where R is a relative signed clique of \((G,\varSigma )\). Given a family \(\mathcal {F}\) of signed graphs, the relative signed clique number is \(\omega _{rs}(\mathcal {F}) = \max \{\omega _{rs}((G,\varSigma ))|(G,\varSigma ) \in \mathcal {F}\}\). For the family \(\mathcal {P}_3\) of signed planar graphs, the problem of finding the value of \(\omega _{rs}(\mathcal {P}_3)\) is an open problem. In this article, we close it by proving \(\omega _{rs}(\mathcal {P}_3)=8\).
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References
Beaudou, L., Foucaud, F., Naserasr, R.: Homomorphism bounds and edge-colourings of k\({}_{\text{4 }}\)-minor-free graphs. J. Comb. Theor. Ser. B 124, 128–164 (2017)
Brewster, R.C., Foucaud, F., Hell, P., Naserasr, R.: The complexity of signed graph and edge-coloured graph homomorphisms. Discret. Math. 340(2), 223–235 (2017)
Das, S., Ghosh, P., Prabhu, S., Sen, S.: Relative clique number of planar signed graphs. Discret. Appl. Math. (accepted)
Harary, F.: On the notion of balance of a signed graph. Mich. Math. J. 2(2), 143–146 (1953)
Naserasr, R., Sen, S., Sun, Q.: Walk-powers and homomorphism bounds of planar signed graphs. Graphs Comb. 32(4), 1505–1519 (2016)
Naserasr, R., Rollová, E., Sopena, É.: Homomorphisms of planar signed graphs to signed projective cubes. Discret. Math. Theor. Comput. Sci. 15(3), 1–12 (2013)
Naserasr, R., Rollová, E., Sopena, É.: Homomorphisms of signed graphs. J. Graph Theor. 79(3), 178–212 (2015)
Ochem, P., Pinlou, A., Sen, S.: Homomorphisms of 2-edge-colored triangle-free planar graphs. J. Graph Theor. 85(1), 258–277 (2017)
Zaslavsky, T.: Characterizations of signed graphs. J. Graph Theor. 25(5), 401–406 (1981)
Zaslavsky, T.: Signed graphs. Discret. Appl. Math. 4(1), 47–74 (1982)
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Das, S., Nandi, S., Sen, S., Seth, R. (2019). The Relative Signed Clique Number of Planar Graphs is 8. In: Pal, S., Vijayakumar, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2019. Lecture Notes in Computer Science(), vol 11394. Springer, Cham. https://doi.org/10.1007/978-3-030-11509-8_20
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DOI: https://doi.org/10.1007/978-3-030-11509-8_20
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