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Equitable Cluster Partition of Planar Graphs with Girth at Least 12
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An equitable(O1k,O2k,...,Omk)-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 Vi,V2,...,Vmsuch that for every integer i in {1,2,...,m},G[Vi]is a graph with components of order at most k,and for each distinct pair i,j in {1,...,m},there is-1＜|Vi|-|Vj|＜1.In this paper,we proved that every planar graph G with minimum degree δ(G)≥ 2 and girth g(G)≥ 12 admits an equitable(O17,O27,...,Om7)-partition,for any integer m ≥ 2.
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