Note on Long Directed Cycles in Eulerian Digraphs

TL;DR

The paper tightens the known lower bound for the length of the longest directed cycle in an $n$-vertex Eulerian digraph from $\sqrt{d}$ to $\sqrt{2d}-\tfrac{3}{2}$ by employing a final out-branching and a level-based arc analysis. The key idea is to replace the DFS-tree approach with a final out-branching, partition vertices into levels, and distinguish forward versus back arcs; this enables a precise arc-count bound: $|A(D)| \le n(t-1) + n t(t+1)/2$, leading to $t \ge \sqrt{2d}-\tfrac{3}{2}$ where $t$ is the cycle length. A central technical point is that in a final out-branching, all arcs from upper to lower levels are back arcs and each level forms an independent set, which tightens the otherwise standard bound. The paper also notes a corollary for vertex-started directed paths and discusses potential broader implications for related long-cycle questions in Eulerian digraphs.

Abstract

Huang, Ma, Shapira, Sudakov and Yuster (Comb. Prob. Comput. 2013) proved that every Eulerian digraph of average out-degree $d$ has a directed cycle of length at least $\sqrt{d}.$ We improve the lower bound from $\sqrt{d}$ to $\sqrt{2d}-3/2.$

Theorems & Definitions (10)

• Conjecture 1.1: Bollobas1996weighted
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['Thm:main']}
• Claim 2.3
• Theorem 3.1