Full Orientability of the Square of a Cycle

TL;DR

This paper proves that the square of a cycle graph C_n is fully orientable for all n except 6, meaning it can realize all intermediate numbers of dependent arcs in acyclic orientations.

Contribution

It establishes the full orientability of the square of cycle graphs C_n for all n except 6, filling a gap in graph orientation theory.

Findings

Square of cycle C_n is fully orientable for all n ≠ 6.

Identifies the exception at n=6 where full orientability does not hold.

Provides a characterization of dependent arcs in acyclic orientations.

Abstract

Let D be an acyclic orientation of a simple graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define m and M to be the minimum and the maximum number of d(D) over all acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n is fully orientable except n=6.