A study on parity signed graphs: the $rna$ number

TL;DR

This paper investigates the properties of parity signed graphs, introduces the concept of the $rna$ number, proves characterizations and bounds for it, and solves related conjectures and problems in the field.

Contribution

It characterizes when the set of negative edge counts is a singleton, establishes the first upper bound for the $rna$ number, and proves an inequality involving graph and complement $rna$ numbers.

Findings

Characterization of graphs with singleton $\Sigma^{-}(G)$ set.

First known upper bound for the $rna$ number.

Proved inequality relating $rna$ numbers of a graph and its complement.

Abstract

The study on parity signed graphs was initiated by Acharya and Kureethara very recently and then followed by Zaslavsky etc.. Let $(G,\sigma)$ be a signed graph on $n$ vertices. If $(G,\sigma)$ is switch-equivalent to $(G,+)$ at a set of $\lfloor \frac{n}{2} \rfloor$ many vertices, then we call $(G,\sigma)$ a parity signed graph and $\sigma$ a parity-signature. $\Sigma^{-}(G)$ is defined as the set of the number of negative edges of $(G,\sigma)$ over all possible parity-signatures $\sigma$. The $rna$ number $\sigma^-(G)$ of $G$ is given by $\sigma^-(G)=\min \Sigma^{-}(G)$. In other words, $\sigma^-(G)$ is the smallest cut size that has nearly equal sides. In this paper, all graphs considered are finite, simple and connected. We apply switch method to the characterization of parity signed graphs and the study on the $rna$ number. We prove that: for any graph $G$, $\Sigma^{-}(G)=\left\{\sigma^{-}(G)\right\}$ if and only if $G$ is $K_{1, n-1} $ with $n$ even or $K_{n}$. This confirms a conjecture proposed in [M. Acharya and J.V. Kureethara. Parity labeling in signed graphs. J. Prime Res. Math., to appear. arXiv:2012.07737]. Moreover, we prove a nontrivial upper bound for the $rna$ number: for any graph $G$ on $m$ edges and $n$ ($n\geq 4$) vertices, $\sigma^{-}(G)\leq \lfloor \frac{m}{2}+\frac{n}{4} \rfloor$. We show that $K_n$, $K_n-e$ and $K_n-\triangle$ are the only three graphs reaching this bound. This is the first upper bound for the $rna$ number so far. Finally, we prove that: for any graph $G$, $\sigma^-(G)+\sigma^-(\overline{G})\leq \sigma^-(G\cup \overline{G})$, where $\overline{G}$ is the complement of $G$. This solves a problem proposed in [M. Acharya, J.V. Kureethara and T. Zaslavsky. Characterizations of some parity signed graphs. 2020, arXiv:2006.03584v3].