Defective Goldberg-Seymour conjecture

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Let d∈N∗d\in\mathbb N^* and let GG be a graph. Define

Γd(G)=max⁡{⌈∣E(G[X])∣⌊d∣X∣/2⌋⌉:X⊆V(G)}.\Gamma_d(G)=\max\left\{\left\lceil \frac{|E(G[X])|}{\left\lfloor d|X|/2\right\rfloor}\right\rceil\mathrel{:}X\subseteq V(G)\right\}.

Defective Goldberg-Seymour conjecture. Every graph GG satisfies

χd′(G)≤max⁡{Γd(G),⌈Δ(G)+1d⌉}.\chi'_d(G)\leq \max\left\{\Gamma_d(G),\left\lceil \frac{\Delta(G)+1}{d}\right\rceil\right\}.

This is proposed as a generalization of the Goldberg-Seymour bound to defective edge colouring; no proof or counterexample is supplied in the source.

References

Primary source

Pierre Aboulker, Guillaume Aubian and Chien-Chung Huang, “Vizing's and Shannon's Theorems for defective edge colouring”, arXiv:2201.11548 (2022).

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