The minimum edge-pancyclic graph of a given order

TL;DR

This work tightens the minimum-size problem for edge-pancyclic graphs by introducing and exploiting the class of $k$-edge-proper graphs to derive stronger lower bounds, showing for large $n$ that $f(n) \ge \lceil \frac{7n}{4} \rceil$. It then provides a gadget-based three-step construction that yields edge-pancyclic graphs with $e(G) \le 2n - \frac{n}{200\ln n}$, improving the known upper bound significantly. Together, these results move $f(n)$ closer to the conjectured asymptotic form and establish a versatile framework (extremal families and gadget replacements) for future tightening.

Abstract

A graph $G$ of order $n$ is called edge-pancyclic if, for every integer $k$ with $3 \leq k \leq n$, every edge of $G$ lies in a cycle of length $k$. Determining the minimum size $f(n)$ of a simple edge-pancyclic graph with $n$ vertices seems difficult. Recently, Li, Liu and Zhan \cite{li2024minimum} gave both a lower bound and an upper bound of $f(n)$. In this paper, we improve their lower bound by considering a new class of graphs and improve the upper bound by constructing a family of edge-pancyclic graphs.

Figures (9)

• Figure 1: The extremal graphs.
• Figure 2: Contract the edge $yv$ as $v'$.
• Figure 3: Two cases about $N(y)\cap N(z)$.
• Figure 4: Three classes of $3$-degree vertices.
• Figure 5: Three types of cycles in Lemma \ref{['lem: each edge in a cycle s^p']}

Theorems & Definitions (12)

• Theorem 1.1: li2024minimum
• Theorem 1.2: li2024minimum
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Conjecture 1.6
• Conjecture 1.7
• Theorem 1.8: li2024minimum
• Theorem 1.9
• Lemma 2.1