On $k$-connected vertex-pancyclic graphs without pancyclic edges

TL;DR

The paper demonstrates that for any positive integer k, there exists a k-connected vertex-pancyclic graph that does not contain any pancyclic edges, answering a question in graph theory.

Contribution

It proves that no universal k exists such that all k-connected vertex-pancyclic graphs contain pancyclic edges.

Findings

For every positive integer k, such graphs exist.

These graphs are k-connected and vertex-pancyclic but lack pancyclic edges.

The result answers a previously posed open question.

Abstract

An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every $2$-connected $[4,2]$-graph of order at least seven contains a pancyclic edge. Zhan asked whether there exists a positive integer $k$ such that every $k$-connected vertex-pancyclic graph contains a pancyclic edge. We answer this question by showing that for every positive integer $k$, there is a $k$-connected vertex-pancyclic graph containing no pancyclic edge.