Neighbour sum distinguishing edge-weightings with local constraints - Archive ouverte HAL Access content directly
Journal Articles Discrete Applied Mathematics Year : 2023

Neighbour sum distinguishing edge-weightings with local constraints

Abstract

A $k$-edge-weighting of $G$ is a mapping $\omega:E(G)\longrightarrow \{1,\ldots,k\}$. The edge-weighting of $G$ naturally induces a vertex-colouring $\sigma_{\omega}:V(G)\longrightarrow \mathbb{N}$ given by $\sigma_{\omega}(v)=\sum_{u\in N_G(v)}\omega(vu)$ for every $v\in V(G)$. The edge-weighting $\omega$ is neighbour sum distinguishing if it yields a proper vertex-colouring $\sigma_{\omega}$, \emph{i.e.}, $\sigma_{\omega}(u)\neq \sigma_{\omega}(v)$ for every edge $uv$ of $G$. We investigate a neighbour sum distinguishing edge-weighting with local constraints, namely, we assume that the set of edges incident to a vertex of large degree is not monochromatic. A graph is nice if it has no components isomorphic to $K_2$. We prove that every nice graph with maximum degree at most~5 admits a neighbour sum distinguishing $(\Delta(G)+2)$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights. Furthermore, we prove that every nice graph admits a neighbour sum distinguishing $7$-edge-weighting such that all the vertices of degree at least~6 are incident with at least two edges of different weights. Finally, we show that nice bipartite graphs admit a neighbour sum distinguishing $6$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights.
Fichier principal
Vignette du fichier
edge-weighting-rev3.pdf (297.57 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03615738 , version 1 (21-03-2022)
hal-03615738 , version 2 (16-01-2023)

Identifiers

Cite

Antoine Dailly, Elżbieta Sidorowicz. Neighbour sum distinguishing edge-weightings with local constraints. Discrete Applied Mathematics, 2023, 336, pp.109-124. ⟨10.1016/j.dam.2023.04.005⟩. ⟨hal-03615738v2⟩
70 View
20 Download

Altmetric

Share

Gmail Facebook X LinkedIn More