划分围长至少为6 的平面图为有界大小的分支

图$G$的$(mathcal{O}_{k_1}, mathcal{O}_{k_2})$-划分是将$V(G)$划分成两个非空子集$V_{1}$和$V_{2}$, 使得$G[V_{1}]$和$G[V_{2}]$分别是分支的阶数至多$k_1$和$k_2$的图.在本文中,我们考虑了有围长限制的平面图的点集划分问题,使得每个部分导出一个具有有界大小分支的图.我们证明了每一个围长至少为6并且$i$-圈不与$j$-圈相交的平面图允许$(mathcal{O}_{2}$, $mathcal{O}_{3})$-划分,其中$iin{6,7,8}$和$jin{6,7,8,9}$.

Partitioning Planar Graphs with Girth at Least $6$ into Bounded Size Components

An ($mathcal{O}_{k_1}$, $mathcal{O}_{k_2}$)-partition of a graph $G$ is the partition of $V(G)$ into two non-empty subsets $V_{1}$ and $V_{2}$, such that $G[V_{1}]$ and $G[V_{2}]$ are graphs with components of order at most $k_1$ and $k_2$, respectively. In this paper, we consider the problem of partitioning the vertex set of a planar graph with girth restriction such that each part induces a graph with components of bounded order. We prove that every planar graph with girth at least $6$ and $i$-cycle is not intersecting with $j$-cycle admits an ($mathcal{O}_{2}$, $mathcal{O}_{3}$)-partition, where $iin{6,7,8}$ and $jin{6,7,8,9}$.