Towards a weighted version of the hajnal-szemerédi theorem

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 Szemerédi 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 ε [0, 1] is given. What is the minimum weighted degree of Kn that guarantees the existence of a K _r -factor such that every factor has total edge weight at least t(^r₂)? We provide some lower and upper bounds and make a conjecture on the asymptotics of the threshold as n goes to infinity.