Palindromic products

Abstract

A graph G on n vertices is palindromic if there is a vertex-labeling bijection f:V(G) --> {1,2, ... ,n} with the property that for any edge  vw of G, there is an edge xy of G for which f(x)=n-f(v)+1 and f(y)=n-f(w)+1.

This notion was defied and explored in a recent paper  [R. Beeler, Palindromic graphs, Bulletin of the ICA, 85 (2019) 85--100]. The paper gives sufficient conditions on the factors of a Cartesian product of graphs that ensure the product is palindromic, but states that it is unknown whether the conditions are necessary. We prove that the conditions are indeed necessary. Further, we prove a parallel result for the strong product of graphs.