Vizing's and Shannon's Theorems for defective edge colouring

TL;DR

This paper extends classical theorems on edge coloring by establishing tight bounds for defective edge coloring in multigraphs and simple graphs, generalizing Vizing and Shannon's results.

Contribution

It provides new bounds for the minimum number of colors needed for defective edge coloring, generalizing and tightening classical theorems for multigraphs and simple graphs.

Findings

For multigraphs, bounds are tight and depend on parity of d.

For simple graphs, the chromatic index is within a small set of values.

Computational complexity results are characterized for certain parameters.

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.