Packing K_rs in bounded degree graphs
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We study the problem of finding a maximum-cardinality set of r-cliques in an undirected graph of fixed maximum degree Δ, subject to the cliques in that set being either vertex-disjoint or edge-disjoint. It is known for r=3 that the vertex-disjoint (edge-disjoint) problem is solvable in linear time if Δ=3 (Δ=4) but APX-hard if Δ≥ 4 (Δ≥ 5). We generalise these results to an arbitrary but fixed r ≥ 3, and provide a complete complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree Δ. Specifically, we show that the vertex-disjoint problem is solvable in linear time if Δ < 3r/2 - 1, solvable in polynomial time if Δ < 5r/3 - 1, and APX-hard if Δ≥⌈ 5r/3 ⌉ - 1. We also show that if r≥ 6 then the above implications also hold for the edge-disjoint problem. If r ≤ 5, then the edge-disjoint problem is solvable in linear time if Δ < 3r/2 - 1, solvable in polynomial time if Δ≤ 2r - 2, and APX-hard if Δ > 2r - 2.
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