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- W2148784152 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." @default.
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- W2148784152 date "2014-10-01" @default.
- W2148784152 modified "2023-09-24" @default.
- W2148784152 title "Vertex arboricity of toroidal graphs with a forbidden cycle" @default.
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- W2148784152 doi "https://doi.org/10.1016/j.disc.2014.06.011" @default.
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