Vizing's And Shannon's Theorems For Defective Edge Colouring - ArXiv

Mathematics > Combinatorics arXiv:2201.11548 (math) [Submitted on 27 Jan 2022 (v1), last revised 3 Feb 2022 (this version, v2)] Title:Vizing's and Shannon's Theorems for defective edge colouring Authors:Pierre Aboulker, Guillaume Aubian, Chien-Chung Huang View a PDF of the paper titled Vizing's and Shannon's Theorems for defective edge colouring, by Pierre Aboulker and Guillaume Aubian and Chien-Chung Huang View PDF

Abstract:We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $\chi'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every integer $d$, every multigraph $G$ with maximum degree $\Delta$ is $(\lceil \frac{\Delta}{d} \rceil, d)$-edge colourable if $d$ is even and $(\lceil \frac{3\Delta - 1}{3d - 1} \rceil, d)$-edge colourable if $d$ is odd and these bounds are tight. We also prove that for every simple graph $G$, $\chi'_{d}(G) \in \{ \lceil \frac{\Delta}{d} \rceil, \lceil \frac{\Delta+1}{d} \rceil \}$ and characterize the values of $d$ and $\Delta$ for which it is NP-complete to compute $\chi'_d(G)$. These results generalize several classic results on the chromatic index of a graph by Shannon, Vizing, Holyer, Leven and Galil.

Subjects:	Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Cite as:	arXiv:2201.11548 [math.CO]
(or arXiv:2201.11548v2 [math.CO] for this version)
https://doi.org/10.48550/arXiv.2201.11548 Focus to learn more arXiv-issued DOI via DataCite

Submission history

From: Pierre Aboulker [view email] [v1] Thu, 27 Jan 2022 14:38:26 UTC (14 KB) [v2] Thu, 3 Feb 2022 20:24:07 UTC (15 KB) Full-text links:

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