Lower bound for sparse k-edge-proper graphs

Establish that every k-edge-proper graph G of sufficiently large order n satisfies e(G)≥(4k−9)n/(2k−4) for each integer k≥4.

Background

A graph is k-edge-proper if every edge lies in a cycle of every length from 3 through k and also lies in a Hamilton cycle. The paper proves the lower bounds e(G)≥5n/3 for 3-edge-proper graphs and e(G)≥7n/4 for 4-edge-proper graphs. It then proposes a general formula for the asymptotic minimum size of k-edge-proper graphs when k≥4.

The conjectured bound is supported by a family constructed from a cycle by replacing each cycle edge with a fan graph F_{2k−3}; this construction is claimed to be k-edge-proper and to attain the proposed asymptotic edge-to-vertex ratio.

References

From the above result, we have the following natural generalization. \begin{conjecture} Let $k\geq 4$ and $n$ large enough. If a graph $G$ of order $n$ is $k$-edge-proper, then $e(G)\geq (4k-9)n/(2k-4)$. \end{conjecture}

— The minimum edge-pancyclic graph of a given order  (2503.05506 - Zhao et al., 7 Mar 2025) in Conjecture immediately following the discussion of the 4-edge-proper extremal construction in Section 1 (Introduction)