Canonical double covers of generalized Petersen graphs, and double generalized Petersen graphs

Abstract

The canonical double cover (Formula presented.) of a graph (Formula presented.) is the direct product of (Formula presented.) and (Formula presented.). If (Formula presented.) then (Formula presented.) is called stable; otherwise (Formula presented.) is called unstable. An unstable graph is said to be nontrivially unstable if it is connected, non-bipartite and no two vertices have the same neighborhood. In 2008 Wilson conjectured that, if the generalized Petersen graph (Formula presented.) is nontrivially unstable, then both (Formula presented.) and (Formula presented.) are even, and either (Formula presented.) is odd and (Formula presented.), or (Formula presented.). In this note we prove t..