Adding direction constraints to the 1-2-3 Conjecture

Abstract

In connection with the so-called 1-2-3 Conjecture, we introduce and study a new variant of proper labellings, obtained when aiming at designing, for an oriented graph, an oriented colouring through the sums of labels incident to its vertices. Formally, for an oriented graph

and a k-labelling

of its arcs, for every vertex

, one can compute the sum $\sigma \left(v\right)$ of labels assigned by ℓ to its incident arcs. We call ℓ an oriented labelling if the sum function σ indeed forms an oriented colouring of

. That is, for any two arcs

and

of

, if $\sigma \left(a\right)=\sigma \left(d\right)$, then we must have $\sigma \left(b\right)\ne \sigma \left(c\right)$. We denote by

the smallest k such that oriented k-labellings of

exist (if any).

We study this new parameter in general and in particular contexts. In particular, we observe that there is no constant bound on

in general, contrarily to the undirected case. Still, we establish connections between this parameter and others, such as the oriented chromatic number, from which we deduce other types of bounds, some of which we improve upon for some classes of oriented graphs. We also investigate other aspects of this parameter, such as the complexity of determining

for a given oriented graph

, or the possible relationships between

and the underlying graph G of

.