A median-type condition for graph tiling

TL;DR

This paper relaxes the minimum degree condition for graph tiling, showing that only a fraction of vertices need to meet the degree requirement for covering a proportion of the graph with disjoint copies of a fixed graph.

Contribution

It introduces a median-type condition for graph tiling, extending previous results by requiring only a fraction of vertices to satisfy the degree condition.

Findings

Minimum degree condition can be relaxed to a fractional vertex condition

Achieves tiling results under less restrictive degree assumptions

Extends classical tiling theorems with median-type conditions

Abstract

Komlos [Tiling Turan theorems, Combinatorica, 20,2 (2000), 203{218] determined the asymptotically optimal minimum degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph. We show that the minimum degree condition can be relaxed in the sense that we require only a given fraction of vertices to have the prescribed degree.