AI 中文总结
本文研究图的k-联盟划分，计算若干图族的k-联盟数，给出不交并和图的联运算下的界，证明其大小构成区间，且每个图都是k-联盟图。
AI 中文摘要
在图中，集合D是k-控制集，当且仅当V(G)\D中的每个顶点在D中至少有k个邻居。Jafari、Alikhani和Bakhshesh引入了k-联盟的概念，它是一对不相交的顶点集X₁和X₂，满足两者都不是k-控制集，但X₁∪X₂是k-控制集。k-联盟划分是一种顶点划分，其中每个集合要么与某个其他集合构成k-联盟，要么本身是恰好包含k个顶点的k-控制集。k-联盟数COₖ(G)是k-联盟划分中集合的最大数量。我们计算了若干图族的k-联盟数，并给出了不交并和图的联运算下k-联盟数的界。我们证明k-联盟划分的可能大小构成一个区间。最后，我们研究了k-联盟图，并证明每个图都是k-联盟图。
英文摘要
In a graph, a set $D$ is $k$-dominating if every vertex in $V(G) \setminus D$ has at least $k$ neighbors in $D$. Jafari, Alikhani, and Bakhshesh introduced the concept of a $k$-coalition, which is a pair of disjoint sets $X_1$ and $X_2$ of vertices such that neither is a $k$-dominating set but $X_1 \cup X_2$ is a $k$-dominating set. A $k$-coalition partition is a vertex partition in which each set either forms a $k$-coalition with some other set or is itself a $k$-dominating set with exactly $k$ vertices. The $k$-coalition number $\operatorname{CO_k}(G)$ is the maximum number of sets in a $k$-coalition partition. We compute the $k$-coalition number for several families and bound the $k$-coalition number under disjoint union and graph join. We show that the set of possible sizes of $k$-coalition partitions forms an interval. Finally, we investigate $k$-coalition graphs and prove that every graph is a $k$-coalition graph.
