k-connectedness and vertex-pancyclicity do not force pancyclic edges

An edge of a graph of order n is pancyclic if it lies in a cycle of every length 3 , … , n . A graph of order n is vertex-pancyclic if every vertex lies in a cycle of every length 3 , … , n . Recently, Li and Zhan proved that every 2-connected graph of order at least seven in which any induced subgraph on four vertices has at least two edges contains a pancyclic edge, and Zhan asked whether there exists a positive integer k such that every k -connected vertex-pancyclic graph contains a pancyclic edge (Li and Zhan 2026). We answer this question by showing that for every positive integer k , there is a k -connected vertex-pancyclic graph containing no pancyclic edge.