Equitable Cluster Partition of Planar Graphs with Girth at Least 12 
Received:March 22, 2023 Revised:December 15, 2023 
Key Words:
equitable cluster partition planar graph girth discharging

Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.12071265; 12271331) and the Natural Science Foundation of Shandong Province (Grant No.ZR202102250232). 

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Abstract: 
An equitable $({\mathcal{O}}^{1}_{k}, {\mathcal{O}}^{2}_{k}, \ldots, {\mathcal{O}}^{m}_{k})$partition of a graph $G$, which is also called a $k$ cluster $m$partition, is the partition of $V(G)$ into $m$ nonempty subsets $V_{1}$, $V_{2}$, \ldots, $V_{m}$ such that for every integer $i$ in $\{1, 2, \ldots, m\}$, $G[V_{i}]$ is a graph with components of order at most $k$, and for each distinct pair $i, j$ in $\{1,\ldots, m\}$, there is $1\leqV_{i}V_{j}\leq1$. In this paper, we proved that every planar graph $G$ with minimum degree $\delta(G)\geq2$ and girth $g(G)\geq12$ admits an equitable $({\mathcal{O}}^{1}_{7}, {\mathcal{O}}^{2}_{7}, \ldots, {\mathcal{O}}^{m}_{7})$partition, for any integer $m\geq2$. 
Citation: 
DOI:10.3770/j.issn:20952651.2024.02.002 
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