A minimum weighted degree above ((1+3t)/4)n suffices to force a perfect packing by K4s of total weight greater than 6t in every large edge-weighted complete graph.
A version of Szemer\'edi's regularity lemma for multicolored graphs and directed graphs that is suitable for induced graphs
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abstract
In this manuscript we develop a version of Szemer\'edi's regularity lemma that is suitable for analyzing multicolorings of complete graphs and directed graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M. Szegedy [Combinatorica, 20(4) (2000), 451--476] who prove a similar result for graphs. The purpose is to extend classical results on dense hereditary properties, such as the speed of the property or edit distance, to the above-mentioned combinatorial objects.
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2025 1verdicts
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Packing tetrahedrons in edge-weighted graphs
A minimum weighted degree above ((1+3t)/4)n suffices to force a perfect packing by K4s of total weight greater than 6t in every large edge-weighted complete graph.