2,470 research outputs found
On Linkedness of Cartesian Product of Graphs
We study linkedness of Cartesian product of graphs and prove that the product
of an -linked and a -linked graphs is -linked if the graphs are
sufficiently large. Further bounds in terms of connectivity are shown. We
determine linkedness of product of paths and product of cycles
Partitioning a graph into highly connected subgraphs
Given , a -proper partition of a graph is a partition
of such that each part of induces a
-connected subgraph of . We prove that if is a graph of order
such that , then has a -proper partition with at
most parts. The bounds on the number of parts and the minimum
degree are both best possible. We then prove that If is a graph of order
with minimum degree , where
, then has a -proper partition into at most
parts. This improves a result of Ferrara, Magnant and
Wenger [Conditions for Families of Disjoint -connected Subgraphs in a Graph,
Discrete Math. 313 (2013), 760--764] and both the degree condition and the
number of parts are best possible up to the constant
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