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On edge-group choosability of graphs
In this paper, we study the concept of edge-group choosability of graphs. We
say that G is edge k-group choosable if its line graph is k-group choosable. An
edge-group choosability version of Vizing conjecture is given. The evidence of
our claim are graphs with maximum degree less than 4, planar graphs with
maximum degree at least 11, planar graphs without small cycles, outerplanar
graphs and near-outerplanar graphs
Vertex Arboricity of Toroidal Graphs with a Forbidden Cycle
The vertex arboricity of a graph is the minimum such that
can be partitioned into sets where each set induces a forest. For a
planar graph , it is known that . In two recent papers, it was
proved that planar graphs without -cycles for some
have vertex arboricity at most 2. For a toroidal graph , it is known that
. Let us consider the following question: do toroidal graphs
without -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
. Since a complete graph on 5 vertices is a toroidal graph without any
-cycles for 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.Comment: 8 pages, 2 figure
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