Asymptotic lower bound for edge-pancyclic graphs

Prove that every edge-pancyclic graph G with sufficiently large order n has at least 2n−o(n) edges.

Background

Every edge-pancyclic graph is k-edge-proper for every k≤n. Consequently, the preceding conjecture for k-edge-proper graphs would imply an asymptotic lower bound approaching 2n for the minimum size f(n) of an edge-pancyclic graph.

The paper establishes the finite lower bound f(n)≥7n/4 for n≥8 and improves a prior upper bound by constructing edge-pancyclic graphs with at most 2n−n/(200 ln n) edges for an infinite family of orders. The precise asymptotic behavior remains unresolved in the stated conjecture.

References

The above conjecture implies the following conjecture. \begin{conjecture} If $G$ is edge-pancyclic with $n$ vertices and $n$ is large enough, then $$e(G)\geq 2n-o(n).$$ \end{conjecture}

— The minimum edge-pancyclic graph of a given order  (2503.05506 - Zhao et al., 7 Mar 2025) in Conjecture immediately following the general k-edge-proper conjecture in Section 1 (Introduction)