1-2-3 Conjecture in digraphs: More results and directions

Abstract

Horňak, Przybyło and Woźniak recently proved that, a small class of obvious exceptions apart, every digraph can be 4-arc-weighted so that, for every arc − → uv, the sum of weights incoming to u is different from the sum of weights outgoing from v. They conjectured a stronger result, namely that the same statement with 3 instead of 4 should also be true. We verify this conjecture in this work.This work takes place in a recent "quest" towards a directed version of the 1-2-3 Conjecture, the variant above being one of the last introduced ones. We take the occasion of this work to establish a summary of all results known in this field, covering known upper bounds, complexity aspects, and choosability. On the way we prove additional results which were missing in the whole picture. We also mention the aspects that remain open.

On Generalisations of the AVD Conjecture to Digraphs

“…Let us mention that other related problems generalised to digraphs were considered before the work of Sopena and Woźniak. In particular, the series [3,4,5,6,11] of works, dedicated to directed variants of the so-called 1-2-3 Conjecture, are very close to the investigations conducted in the current paper. It turns out, actually, that some of the phenomena observed through our results and the proof techniques we develop, are reminiscent of some from these works.…”

“…Using different arguments, for every (−, +)-nice digraph D, we provide an upper bound on ndi −,+ (D) that is linear in ∆ * (D). Just as in Section 3, this is by exploiting some relationship between an arc-colouring of D and an edge-colouring of B(D), the bipartite graph associated to D. However, as highlighted earlier in [5], note that using the relationship is not so natural in the present context. Indeed, by a proper edge-colouring of B(D), having S(u + ) = S(v − ) is not so relevant regarding D, as, when transposing the edge-colouring to an arc-colouring of D, this would yield S + (u) = S − (v), which is not required in the (−, +) version.…”

“…Note that the situation remains unchanged if we add the arc ts to D, since, here, we would always get S − (s) = α = S + (t), when assigning a colour α to ts. If D has two such adjacent vertices, we say that D has a bad configuration (such configurations were already considered in [5,11]). Again, in this variant, a (−, +)-nice digraph is a digraph D for which ndi −,+ (D) is defined.…”

Approaches that output infinitely many graphs with small local antimagic chromatic number

“…Example 2.1. For p = 11, t = 2, using the path P 1,2 with edge set (1,3), (1,4), (2,3), (2,4), (3,4)}}. The graph L 2 (11, 2|1, 1, 4)+ L 2 (11, 2|2, 1, 3) is given in Figure 3.…”

“…Moreover, f is called a local antimagic c(f )-coloring and G is local antimagic c(f )-colorable. The local antimagic chromatic number χ la (G) is defined to be the minimum number of colors taken over all colorings of G induced by local antimatic labelings of G (see [1,3]). Affirmative solutions on some problems raised in [1] can be found in [6].…”