Towards a weighted version of the Hajnal-Szemer\'edi Theorem
Abstract
For a positive integer r>=2, a K_r-factor of a graph is a collection vertex-disjoint copies of K_r which covers all the vertices of the given graph. The celebrated theorem of Hajnal and Szemer\'edi asserts that every graph on n vertices with minimum degree at least (1-1/r)n contains a K_r-factor. In this note, we propose investigating the relation between minimum degree and existence of perfect K_r-packing for edge-weighted graphs. The main question we study is the following. Suppose that a positive integer r>=2 and a real t in [0,1] is given. What is the minimum weighted degree of K_n that guarantees the existence of a K_r-factor such that every factor has total edge weight at least tr(r-1)/2? We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as n goes to infinity.
Keywords
Cite
@article{arxiv.1206.1376,
title = {Towards a weighted version of the Hajnal-Szemer\'edi Theorem},
author = {József Balogh and Graeme Kemkes and Choongbum Lee and Stephen J. Young},
journal= {arXiv preprint arXiv:1206.1376},
year = {2013}
}
Comments
10 pages; This is the long version of a "problem paper" in Combinatorics, Probability and Computing. In Version 2 we added more details and corrected typographical errors