Given an undirected graph, in the AVD (edge-colouring) Conjecture, the goal is to find a proper edge-colouring with the least number of colours such that every two adjacent vertices are incident to different sets of colours. More precisely, the conjecture says that, a few exceptions apart, every graph G should admit such an edge-colouring with at most \(\Delta (G)+2\) colours. Several aspects of interest behind this problem have been investigated over the recent years, including verifications of the conjecture for particular graph classes, general approximations of the conjecture, and multiple generalisations. In this paper, following a recent work of Sopena and Woźniak, generalisations of the AVD Conjecture to digraphs are investigated. More precisely, four of the several possible ways of generalising the conjecture are focused upon. We completely settle one of our four variants, while, for the three remaining ones, we provide partial results.

给定一个无向图，在AVD（边缘着色）猜想中，目标是找到具有最少颜色数量的适当边缘着色，以使每两个相邻的顶点入射到不同的颜色集。更准确地说，该猜想表明，除少数例外，每个图G都应允许这样的边缘着色，其最多为\（\ Delta（G）+2 \）颜色。近年来，已经研究了这个问题背后令人感兴趣的几个方面，包括对特定图类的猜想的验证，猜想的一般近似以及多种概括。本文根据Sopena和Woźniak的最新工作，研究了AVD猜想对有向图的推广。更准确地说，着重于推测该推测的几种可能方式中的四种。我们完全解决了四个变体之一，而对于其余三个变体，我们提供了部分结果。