The vertex-colouring {a,b}-edge-weighting problem is NP-complete for every pair of weights
type: Dataset
description: Let G be a graph. From an edge-weighting w : E(G) -> {a,b} of G such that a and b are two distinct real numbers, one obtains a vertex-colouring chi_w of G defined as chi_w(u) = sum_{v in N(u)} w(uv) for every u in V(G). If chi_w is a proper colouring of G, i.e. two adjacent vertices of G receive distinct colours by chi_w, then we say that w is vertex-colouring. We investigate the complexity of the problem of deciding whether a graph admits a vertex-colouring edge-weighting taking values among a given pair {a,b}, which is already known to be NP-complete when {a,b} is either {0,1} or {1,2}. We show this problem to be NP-complete for every pair of real weights.
landing page: https://hal.science/hal-00826346
keyword: 1-2-3-conjecture
keyword: complexity
keyword: infoeu-reposemanticsreport
keyword: infoinfo-dmcomputer-science-csdiscrete-mathematics-csdm
keyword: reports
keyword: two-weights
keyword: vertex-colouring-edge-weighting-of-graphs
issued: 2026-05-11T00:20:15.306682
modified: 2026-05-11T00:20:15.306686
language: en
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

The vertex-colouring {a,b}-edge-weighting problem is NP-complete for every pair of weights
type: Distribution
format: HTML
access URL: https://hal.science/hal-00826346
issued: 2026-05-11T00:20:15.316919
modified: 2026-05-11T00:20:15.298058