  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.

Combinatorics