The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture

Abstract

Let k be an odd natural number ≥ 5, and let G be a (6k − 7)-edge-connected graph of bipartite index at least k − 1. Then, for each mapping f : V (G) → N, G has a subgraph H such that each vertex v has H-degree f (v) modulo k. We apply this to prove that, if c : V (G) → Z k is a proper vertex-coloring of a graph G of chromatic number k ≥ 5 or k − 1 ≥ 6, then each edge of G can be assigned a weight 1 or 2 such that each weighted vertex-degree of G is congruent to c modulo k. Consequently, each nonbipartite (6k − 7)-edge-connected graph of chromatic number at most k (where k is any odd natural number ≥ 3) has an edge-weighting with weights 1, 2 such that neighboring vertices have distinct weighted degrees (even after reducing these weighted degrees modulo k). We characterize completely the bipartite graph having an edge-weighting with weights 1, 2 such that neighboring vertices have distinct weighted degrees. In particular, that problem belongs to P while it is NP-complete for nonbipartite graphs. The characterization also implies that every 3-edge-connected bipartite graph with at least 3 vertices has such an edge-labelling, and so does every simple bipartite graph of minimum degree at least 3.

Neighbour-sum-2-distinguishing edge-weightings: Doubling the 1–2–3 Conjecture

“…proved by Thomassen, Wu and Zhang [7] that a bipartite graph G satisfies χ Σ (G) = 3 if and only if G is an odd multicactus. Odd multicacti can be defined as follows.…”

“…It is worth recalling that determining the value of χ Σ (G) for a given graph G is NPcomplete in general (Dudek and Wajc [1]), but can be done in polynomial time when restricted to bipartite graphs (Thomassen, Wu and Yang [7]). Hence our result in Subsection 4.1 shows another difference between the parameters χ Σ and χ Σ>1 .…”

“…More generally, it is NP-complete to decide whether χ Σ (G) ≤ 2 holds for a given graph G, as first proved by Dudek and Wajc [1]. The same problem, however, can be handled in polynomial time when restricted to bipartite graphs, as recently shown by Thomassen, Wu and Zhang [7]. To date, the best result towards the 1-2-3 Conjecture is due to Kalkowski, Karoński and Pfender [3], who proved that χ Σ (G) ≤ 5 holds for every nice graph G.…”

The 1–2–3 Conjecture almost holds for regular graphs

“…[5,11,16] for results concerning list versions of the both problems, and in particular very nice result of [15], obtained via algebraic approach exploiting so-called Combinatorial Nullstellensatz due to Alon [4]. Apart from these it is known that 1-2-3 Conjecture holds for 3-colourable graphs, see [9], and [12,13] for other results. In this paper we further develop the techniques from [7] and [6] to show that the weight set {1, 2, 3, 4, 5} from [8] can be narrowed down to presumably almost optimal {1, 2, 3, 4} in the case of regular graphs, what was earlier known for 5-regular graphs [6] and possibly 4-regular ones -see e.g.…”

Note on the group edge irregularity strength of graphs

“…The smallest value of s that allows an irregular labeling is called the irregularity strength of G and denoted by s(G). This problem was one of the major sources of inspiration in graph theory [3,4,5,6,7,12,18,19,20,23,26,28]. For example the concept of G-irregular labeling is a generalization of irregular labeling on Abelian groups.…”