RNA Number of Some Parity Signed Generalized Petersen Graphs

TL;DR

This paper investigates the rna number of parity signed generalized Petersen graphs, establishing bounds, exact values for specific cases, and proposing an algorithm for general graphs.

Contribution

It provides bounds and exact values for the rna number of certain generalized Petersen graphs and introduces an algorithm for computing the rna number of any simple connected graph.

Findings

Bounds for rna number of generalized Petersen graphs are established.

Exact rna numbers are determined for specific Petersen graphs.

An algorithm with exponential and polynomial components is proposed for general graphs.

Abstract

A signed graph $\Sigma=(G,\sigma)$ is said to be parity signed if there exists a bijection $f : V(G) \rightarrow \{1,2,...,|V(G)|\}$ such that $\sigma(uv)=+$ if and only if $f(u)$ and $f(v)$ are of same parity, where $uv$ is an edge of $G$. The rna number of a graph $G$, denoted $\sigma^{-}(G)$, is the minimum number of negative edges among all possible parity signed graphs over $G$. The rna number is also equal to the minimum cut size that has nearly equal sides. In this paper, for generalized Petersen graph $P(n,k)$, we prove that $3 \leq \sigma^{-}(P(n,k)) \leq n$ and these bounds are sharp. The exact value of $\sigma^{-}(P(n,k))$ is determined for $k=1,2$. Some famous generalized Petersen graphs namely, Petersen graph $P(5,2)$, Durer graph $P(6,2)$, Mobius-Kantor graph $P(8,3)$, Dodecahedron $P(10,2)$, Desargues graph $P(10,3)$ and Nauru graph $P(12,5)$ are also treated. We show that the minimum order of a $(4n-1)$-regular graph having rna number one is bounded above by $12n-2$. The sharpness of this upper bound is also shown for $n=1$. We also show that the minimum order of a $(4n+1)$-regular graph having rna number one is $8n+6$. Finally, for any simple connected graph of order $n$, we propose an $O(2^n + n^{\lfloor \frac{n}{2} \rfloor})$ time algorithm for computing its rna number.