The Relative Signed Clique Number of Planar Graphs is 8

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 $\omega _{rs}(\mathcal {P}_3)=8$.