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Article Dans Une Revue Applied Mathematics and Computation Année : 2022

The Neighbour Sum Distinguishing Relaxed Edge Colouring

Antoine Dailly
Eric Duchene
Aline Parreau

Résumé

A d-relaxed k-edge colouring is an edge colouring using colours from the set {1,. .. , k} such that each monochromatic set of edges induces a subgraph with maximum degree at most d. A neighbour sum distinguishing d-relaxed k-edge colouring of G is a d-relaxed k-edge colouring such that for each edge uv ∈ E(G), the sum of colours taken on the edges incident to u is different from the sum of colours taken on the edges incident to v. By χ d (G) we denote the smallest value k in such a colouring of G. In this paper, we prove that χ 2 (G) ≤ 4 for every connected subcubic graph with at least three vertices. For complete graphs with at least three vertices, we show that χ d (K n) ≤ 4 if d ∈ { (n − 1)/2 ,. .. , n − 1} and we also determine the exact value of χ 2 (K n). Finally, we determine the value of χ d (T) for any tree T with at least three vertices.
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Dates et versions

hal-03064954 , version 1 (14-12-2020)

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Antoine Dailly, Eric Duchene, Aline Parreau, Elżbieta Sidorowicz. The Neighbour Sum Distinguishing Relaxed Edge Colouring. Applied Mathematics and Computation, 2022, 419, pp.126864. ⟨10.1016/j.amc.2021.126864⟩. ⟨hal-03064954⟩
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