We study the problem of finding a maximum-cardinality set of $r$-cliques in an undirected graph of fixed maximum degree $\Delta$, 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 $\Delta=3$ ($\Delta=4$) but APX-hard if $\Delta \geq 4$ ($\Delta \geq 5$). We generalise these results to an arbitrary but fixed $r \geq 3$, and provide a complete complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree $\Delta$. Specifically, we show that the vertex-disjoint problem is solvable in linear time if $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta < 5r/3 - 1$, and APX-hard if $\Delta \geq \lceil 5r/3 \rceil - 1$. We also show that if $r\geq 6$ then the above implications also hold for the edge-disjoint problem. If $r \leq 5$, then the edge-disjoint problem is solvable in linear time if $\Delta < 3r/2 - 1$, solvable in polynomial time if $\Delta \leq 2r - 2$, and APX-hard if $\Delta > 2r - 2$.

翻译：我们研究在固定最大度$\Delta$的无向图中，寻找最大基数$r$-团集合的问题，且该集合中的团需满足顶点不交或边不交。已知当$r=3$时，若$\Delta=3$（$\Delta=4$），顶点不交（边不交）问题可在线性时间内求解，但若$\Delta \geq 4$（$\Delta \geq 5$），则问题为APX难。我们将这些结果推广到任意但固定的$r \geq 3$，并给出了最大度$\Delta$图中顶点不交和边不交变体问题的完整复杂度分类。具体而言，我们证明顶点不交问题在$\Delta < 3r/2 - 1$时可在线性时间内求解，在$\Delta < 5r/3 - 1$时可在多项式时间内求解，而在$\Delta \geq \lceil 5r/3 \rceil - 1$时为APX难。我们还证明，若$r\geq 6$，则上述结论同样适用于边不交问题。若$r \leq 5$，则边不交问题在$\Delta < 3r/2 - 1$时可在线性时间内求解，在$\Delta \leq 2r - 2$时可在多项式时间内求解，而在$\Delta > 2r - 2$时为APX难。