13 research outputs found
Acyclic 4-choosability of planar graphs without 4-cycles
summary:A proper vertex coloring of a graph is acyclic if there is no bicolored cycle in . In other words, each cycle of must be colored with at least three colors. Given a list assignment , if there exists an acyclic coloring of such that for all , then we say that is acyclically -colorable. If is acyclically -colorable for any list assignment with for all , then is acyclically -choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles is acyclically 4-choosable. However, this has been as yet verified only for some restricted classes of planar graphs. In this paper, we prove that every planar graph with neither 4-cycles nor intersecting -cycles for each is acyclically 4-choosable
Equitable partition of planar graphs
An equitable -partition of a graph is a collection of induced
subgraphs of such that
is a partition of and
for all . We prove that every planar graph admits an equitable
-partition into -degenerate graphs, an equitable -partition into
-degenerate graphs, and an equitable -partition into two forests and one
graph.Comment: 12 pages; revised; accepted to Discrete Mat
Acyclic 5-Choosability of Planar Graphs Without Adjacent Short Cycles
The conjecture claiming that every planar graph is acyclic 5-choosable[Borodin et al., 2002] has been verified for several restricted classes of planargraphs. Recently, O. V. Borodin and A. O. Ivanova, [Journal of Graph Theory,68(2), October 2011, 169-176], have shown that a planar graph is acyclically 5-choosable if it does not contain an i-cycle adjacent to a j-cycle, where 3<=j<=5 if i=3 and 4<=j<=6 if i=4. We improve the above mentioned result and prove that every planar graph without an i-cycle adjacent to a j-cycle with3<=j<=5 if i=3 and 4<=j<=5 if i=4 is acyclically 5-choosable
Acyclic 4-choosability of planar graphs without cycles of specific length
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