标题： 关于连通联盟数 显示英文标题

标题： On the connected coalition number

摘要： 对于图$G=(V,E)$，顶点不相交的集合$A_{1}$和$A_{2}$形成$G$的连通联盟，如果$A_{1}\cup A_{2}$是一个连通支配集，但$A_{1}$和$A_{2}$都不是连通支配集。 一个连通联盟划分 of$G$是一个划分$\Phi$of$V(G)$，使得$\Phi$中的每个集合要么仅由一个度数为$|V(G)|-1$的顶点组成，要么与$\Phi$中的另一个集合形成$G$的连通联盟。 连通联盟数 of $G$，记作 $CC(G)$，是 $G$的连通联盟划分的最大可能大小。 在本文中，我们刻画满足 $CC(G)=2$的图。 此外，我们得到了单环图以及两个图的冠积和并接的连通联盟数。 最后，我们给出了两个图的笛卡尔积和字典序积的连通联盟数的一个下界。 显示英文摘要

摘要： For a graph $G=(V,E)$, a pair of vertex disjoint sets $A_{1}$ and $A_{2}$ form a connected coalition of $G$, if $A_{1}\cup A_{2}$ is a connected dominating set, but neither $A_{1}$ nor $A_{2}$ is a connected dominating set. A connected coalition partition of $G$ is a partition $\Phi$ of $V(G)$ such that each set in $\Phi$ either consists of only a singe vertex with the degree $|V(G)|-1$, or forms a connected coalition of $G$ with another set in $\Phi$. The connected coalition number of $G$, denoted by $CC(G)$, is the largest possible size of a connected coalition partition of $G$. In this paper, we characterize graphs that satisfy $CC(G)=2$. Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs.