On the rna number of powers of cycles

TL;DR

This paper investigates the rna number of powers of cycles, establishing bounds and exact values for specific cases, thereby extending understanding of parity signed graphs and their minimal negative edge configurations.

Contribution

It provides new bounds for the rna number of cycle powers and determines exact values for the cases when d=2 and d=3.

Findings

The rna number of $C_n^{d}$ is at least 2d.

The rna number of $C_n^{d}$ is at most d(d+1).

For $d=2$ and $d=3$, the rna number equals the upper bound d(d+1).

Abstract

A signed graph $(G,\sigma)$ on $n$ vertices is called a \textit{parity signed graph} if there is a bijective mapping $f \colon V(G) \rightarrow \{1,\ldots,n\}$ such that $f(u)$ and $f(v)$ have same parity if $\sigma(uv)=1$, and opposite parities if $\sigma(uv)=-1$ for each edge $uv$ in $G$. The \emph{rna} number $\sigma^{-}(G)$ of $G$ is the least number of negative edges among all possible parity signed graphs over $G$. In other words, $\sigma^{-}(G)$ is the smallest size of an edge-cut of $G$ such that the sizes of two sides differ at most one. Let $C_n^{d}$ be the $d\text{th}$ power of a cycle of order $n$. Recently, Acharya, Kureethara and Zaslavsky proved that the \emph{rna} number of a cycle $C_n$ on $n$ vertices is $2$. In this paper, we show for $2 \leq d < \lfloor \frac{n}{2} \rfloor$ that $2d \leq \sigma^{-}(C_n^{d}) \leq d(d+1)$. Moreover, we prove that the graphs $C_n^{2}$ and $C_n^{3}$ achieve the upper bound of $d(d+1)$.