Neighbour sum distinguishing edge-weightings with local constraints

TL;DR

This paper studies neighbor sum distinguishing edge-weightings with local constraints, providing new bounds for various classes of graphs, including bipartite graphs, with specific weightings ensuring vertex color distinctions.

Contribution

It introduces new bounds for neighbor sum distinguishing edge-weightings with local constraints for different graph classes, including maximum degree and bipartite graphs.

Findings

Graphs with maximum degree at most 5 admit a neighbor sum distinguishing (Δ(G)+2)-edge-weighting.

Nice graphs admit a neighbor sum distinguishing 7-edge-weighting with degree at least 6 vertices incident to edges of different weights.

Nice bipartite graphs admit a 6-edge-weighting with vertices of degree at least 2 incident to edges of different weights.

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.