Edge Cover Colouring Versus Minimum Degree in Multigraphs
Abstract
An edge colouring of a multigraph can be thought of as a partition of the edges into matchings (a matching meets each vertex at most once). Analogously, an edge cover colouring is a partition of the edges into edge covers (an edge cover meets each vertex at least once). We aim to determine a tight lower bound on the maximum number of parts in an edge cover colouring as a function of the minimum degree delta, which would be an analogue of Shannon's theorem from 1949 on edge-colouring multigraphs. We are able to give a lower bound that is tight except when delta=9 or delta is odd and > 12; in these non-tight cases the best upper and lower bounds differ by one.
Keywords
Cite
@article{arxiv.0906.1778,
title = {Edge Cover Colouring Versus Minimum Degree in Multigraphs},
author = {David Pritchard},
journal= {arXiv preprint arXiv:0906.1778},
year = {2010}
}
Comments
This paper has been withdrawn by the author. The main open question here was proved in [R. P. Gupta. On the chromatic index and the cover index of a mulltigraph. In Th. & Appl. of Graphs: Int. Conf. Kalamazoo, May 11-15, 1978, volume 642/1978 of Lecture Notes in Mathematics, pages 204-215. Springer Verlag, 1976.]. See also a very slick proof in [N. Alon, R. Berke, K. Buchin, M. Buchin, P. Csorba, S. Shannigrahi, B. Speckmann, and P. Zumstein. Polychromatic colorings of plane graphs. Discrete & Computational Geometry, 42(3):421-442, 2009. Preliminary version appeared in Proc. 24th SOCG, pages 338-345, 2008.]