Equitable Cluster Partition of Planar Graphs with Girth at Least 12

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 $m$ non-empty 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 $-1\leq|V_{i}|-|V_{j}|\leq1$. 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 $m\geq2$.