The Point-Arboricity of Planar Graphs

Equitable list point arboricity of graphs

“…A graph G is claw-free if any 3-vertex subgraph of G can not induce a K 1,3 . Note that any graph obtained from a claw-free graph by removing some vertices is also claw-free.…”

“…For any undefined notions we refer the readers to [1]. The point arboricity, or vertex arboricity of G, which is introduced by Chartrand et al [3] and denoted by ρ(G), is the minimum number of colors that can be used to color the vertices of G so that each color class induces an acyclic subgraph of G. In 2000, Borodin, Kostochka and Toft [2] introduced the list version of point arboricity. A graph G is list point k-arborable if, whenever we are given a k-list assignment L(v) of colors for each vertex v ∈ V(G), we can choose a color c(v) ∈ L(v) for each vertex v so that each color class induces an acyclic subgraph of G.…”

Conditional resolvability in graphs: a survey

“…For a connected graph G, a k-partition Π = {S 1 ,S 2 ,...,S k } of V (G) is acyclic if the subgraph S i induced by S i is acyclic in G for each i (1 ≤ i ≤ k). The vertex-arboricity a(G) of G is defined in [11,12] as the minimum k such that V (G) has an acyclic k-partition. If an acyclic partition Π of V (G) is also a resolving partition, then Π is called a resolving acyclic partition of G, which was introduced and studied in [48,49].…”

A Structural Property of Trees with an Application to Vertex‐Arboricity

“…Among other things, they proved that the vertex-arboricity of planar graphs is at most 3. Chartrand and Kronk [2] showed that this bound is sharp by presenting a planar graph of the vertex-arboricity 3. More generally, Kronk [3] showed that if is a surface with Euler genus , then ( ) = 3 and if is the sphere or the Klein bottle, then ( ) = ⌊(9 + √1 + 24 )/4⌋.…”