No abstract

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 following question: do toroidal graphs without k-cycles have vertex arboricity at most 2? It was known that the question is true for k = 3, and recently, Zhang proved the question is true for k = 5. Since a complete graph on 5 vertices is a toroidal graph without any k-cycles for k ≥ 6 and has vertex arboricity at least three, the only unknown case was k = 4. We solve this case in the affirmative; namely, we show that toroidal graphs without 4-cycles have vertex arboricity at most 2.

The choice number of a graph G,denoted by Z,(G),is the minimum number k such that if a list of k colors is given to each vertex of G,there is a vertex coloring of G where each vertex receives a color from its own list no matter what the lists are. In this paper,it is showed that Z,(G)~3 for each plane graph of girth not less than 4 which contains no 6-,7-and 9cycles.