the vertex arboricity $rho(g)$ of a graph $g$ is the minimum number of subsets into which the vertex set $v(g)$ can be partitioned so that each subset induces an acyclic graph‎. ‎a graph $g$ is called list vertex $k$-arborable if for any set $l(v)$ of cardinality at least $k$ at each vertex $v$ of $g$‎, ‎one can choose a color for each $v$ from its list $l(v)$ so that the subgraph induced by ev...

The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by ev...

The vertex arboricity a(G) of a graph G is the minimum k such that V (G) can be partitioned into k sets where each set induces a forest. For a planar graph G, it is known that a(G) ≤ 3. In two recent papers, it was proved that planar graphs without k-cycles for some k ∈ {3, 4, 5, 6, 7} have vertex arboricity at most 2. For a toroidal graph G, it is known that a(G) ≤ 4. Let us consider the follo...

A (d, k) -forest is a forest consisting of trees whose diameters are at most d and whose maximum vertex degree ,6. is at most k. The (d, k)-arboricity of a graph G is the minimum number of (d, k}-forests needed to cover E(G). This concept is a common generalization of linear k-arboricity and star arboricity. Using a probabilistic approach developed recently for linear karboricity, we obtain an ...

The linear vertex-arboricity of a graph G is defined to the minimum number of subsets into which the vertex-set G can be partitioned so that every subset induces a linear forest. In this paper, we give the upper and lower bounds for sum and product of linear vertex-arboricity with independence number and with clique cover number respectively. All of these bounds are sharp.

An integer distance graph is a graph G(D) with the set Z of all integers as vertex set and two vertices u, v ∈ Z are adjacent if and only if |u− v| ∈ D, where the distance set D is a subset of positive integers. A k-vertex coloring of a graph G is a mapping f from V (G) to [0, k − 1]. A path k-vertex coloring of a graph G is a k-vertex coloring such that every connected component is a path in t...