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).
Author NameAffiliation
Xiaoling LIU Department of Mathematics and Statistics, Shandong Normal University, Shandong 250358, P. R. China 
Lei SUN Department of Mathematics and Statistics, Shandong Normal University, Shandong 250358, P. R. China 
Wei ZHENG Department of Mathematics and Statistics, Shandong Normal University, Shandong 250358, P. R. China 
Hits: 101
Download times: 160
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 $m\geq2$.
Citation:
DOI:10.3770/j.issn:2095-2651.2024.02.002
View Full Text  View/Add Comment