The strong equitable vertex 2-arboricity of complete bipartite and tripartite graphs

Abstract

A $(q,r)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r.$ An \emph{equitable $(q,r)$-tree-coloring} of a graph $G$ is a $(q,r)$-tree-coloring such that the sizes of any two color classes differ by at most one. Let the \emph{strong equitable vertex $r$-arboricity} be the minimum $p$ such that $G$ has an equitable $(q,r)$-tree-coloring for every $q\ge p.$ In this paper, we find the exact value for each $v{a}_2^{\equiv }(K_{m,n})$ and $v{a}_2^{\equiv }(K_{l,m,n}).$