# Maximum Frustration in Signed Generalized Petersen Graphs

## TL;DR

This paper investigates the maximum frustration index in signed generalized Petersen graphs, establishing upper bounds for different cases of the graph parameters and demonstrating when these bounds are tight.

## Contribution

It provides new bounds on the maximum frustration index in signed generalized Petersen graphs, extending understanding of their structural properties.

## Key findings

- Maximum frustration of $P_{n,k}$ with $ ext{gcd}(n,k)=1$ is at most $loor{rac{n}{2}} + 1$.
- The bound is tight for $k=1,2,3$.
- For $ ext{gcd}(n,k)=d extgreater 1$, the maximum frustration is bounded by $dloor{rac{n}{2d}} + d + 1$.

## Abstract

A \textit{signed graph} is a simple graph whose edges are labelled with positive or negative signs. A cycle is \textit{positive} if the product of its edge signs is positive. A signed graph is \textit{balanced} if every cycle in the graph is positive. The \textit{frustration index} of a signed graph is the minimum number of edges whose deletion makes the graph balanced. The \textit{maximum frustration} of a graph is the maximum frustration index over all sign labellings. In this paper, first, we prove that the maximum frustration of generalized Petersen graphs $P_{n,k}$ is bounded above by $\left\lfloor \frac{n}{2} \right\rfloor + 1$ for $\gcd(n,k)=1$, and this bound is achieved for $k=1,2,3$. Second, we prove that the maximum frustration of $P_{n,k}$ is bounded above by $d\left\lfloor \frac{n}{2d} \right\rfloor + d + 1$, where $\gcd(n,k)=d\geq2$.