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Packing chromatic vertex-critical graphs
The packing chromatic number of a graph is the smallest
integer such that the vertex set of can be partitioned into sets ,
, where vertices in are pairwise at distance at least .
Packing chromatic vertex-critical graphs, -critical for short, are
introduced as the graphs for which
holds for every vertex of . If , then is
--critical. It is shown that if is -critical,
then the set can be almost
arbitrary. The --critical graphs are characterized, and
--critical graphs are characterized in the case when they
contain a cycle of length at least which is not congruent to modulo
. It is shown that for every integer there exists a
--critical tree and that a --critical
caterpillar exists if and only if . Cartesian products are also
considered and in particular it is proved that if and are
vertex-transitive graphs and , then is -critical
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