Equitable vertex arboricity of graphs

Abstract

An equitable $(t,k)$-tree-coloring of a graph $G$ is a coloring of 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 $k$. The minimum $t$ such that $G$ has an equitable $(t^{\prime },k)$-tree-coloring for every $t^{\prime }\ge t$, denoted by $v{a}_k^{\equiv }\left(G\right)$, is the strong equitable vertex $k$-arboricity. In this paper, we give sharp upper bounds for $v{a}_1^{\equiv }\left(K_{n,n}\right)$ and $v{a}_k^{\equiv }\left(K_{n,n}\right)$, and prove that $v{a}_{\infty }^{\equiv }\left(G\right)\le 3$ for every planar graph $G$ with girth at least 5 and $v{a}_{\infty }^{\equiv }\left(G\right)\le 2$ for every planar graph $G$ with girth at least 6 and for every outerplanar graph.