Partitioning planar graphs without 4-cycles and 6-cycles into a forest and a disjoint union of paths
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the v...
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Zusammenfassung: | In this paper, we show that every planar graph without $4$-cycles and
$6$-cycles has a partition of its vertex set into two sets, where one set
induces a forest, and the other induces a forest with maximum degree at most
$2$ (equivalently, a disjoint union of paths).
Note that we can partition the vertex set of a forest into two independent
sets. However, a pair of independent sets combined may not induce a forest.
Thus our result extends the result of Wang and Xu (2013) stating that the
vertex set of every planar graph without $4$-cycles and $6$-cycles can be
partitioned into three sets, where one induces a graph with maximum degree two,
and the remaining two are independent sets. |
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DOI: | 10.48550/arxiv.2203.06466 |