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- W2002256912 abstract "An equitable $(t,k,d)$-tree-coloring of a graph $G$ is a coloring to vertices of $G$ such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most $k$ and diameter at most $d$. The minimum $t$ such that $G$ has an equitable $(t',k,d)$-tree-coloring for every $t'geq t$ is called the strong equitable $(k,d)$-vertex-arboricity and denoted by $va^{equiv}_{k,d}(G)$. In this paper, we give sharp upper bounds for $va^{equiv}_{1,1}(K_{n,n})$ and $va^{equiv}_{k,infty}(K_{n,n})$ by showing that $va^{equiv}_{1,1}(K_{n,n})=O(n)$ and $va^{equiv}_{k,infty}(K_{n,n})=O(n^{1/2})$ for every $kgeq 2$. It is also proved that $va^{equiv}_{infty,infty}(G)leq 3$ for every planar graph $G$ with girth at least 5 and $va^{equiv}_{infty,infty}(G)leq 2$ for every planar graph $G$ with girth at least 6 and for every outerplanar graph. We conjecture that $va^{equiv}_{infty,infty}(G)=O(1)$ for every planar graph and $va^{equiv}_{infty,infty}(G)leq lceilfrac{Delta(G)+1}{2}rceil$ for every graph $G$." @default.
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- W2002256912 date "2012-11-17" @default.
- W2002256912 modified "2023-09-27" @default.
- W2002256912 title "Equitable vertex arboricity of graphs" @default.
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- W2002256912 doi "https://doi.org/10.48550/arxiv.1211.4193" @default.
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