Maximum Frustration in Signed Generalized Petersen Graphs

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 $\lfloor \frac{n}{2}\rfloor +1$ 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 $d\lfloor \frac{n}{2d}\rfloor +d+1$, where $\gcd (n,k)=d\ge 2$.

References in corpus (1)

• Signed Chromatic Polynomials of Signed Book Graphs