4,379 research outputs found
Nonrepetitive colorings of lexicographic product of graphs
A coloring of the vertices of a graph is nonrepetitive if there
exists no path for which for all
. Given graphs and with , the lexicographic
product is the graph obtained by substituting every vertex of by a
copy of , and every edge of by a copy of . %Our main results
are the following. We prove that for a sufficiently long path , a
nonrepetitive coloring of needs at least
colors. If then we need exactly colors to nonrepetitively color
, where is the empty graph on vertices. If we further require
that every copy of be rainbow-colored and the path is sufficiently
long, then the smallest number of colors needed for is at least
and at most . Finally, we define fractional nonrepetitive
colorings of graphs and consider the connections between this notion and the
above results
Nonrepetitive colorings of lexicographic product of paths and other graphs
A coloring of the vertices of a graph is nonrepetitive if
there exists no path for which
for all . Given graphs and
with , the lexicographic product is the graph
obtained by substituting every vertex of by a copy of , and
every edge of by a copy of .
We prove that for a sufficiently long path , a nonrepetitive
coloring of needs at least
colors. If then we need exactly colors to
nonrepetitively color , where is the empty graph on
vertices. If we further require that every copy of be
rainbow-colored and the path is sufficiently long, then the
smallest number of colors needed for is at least and
at most . Finally, we define fractional
nonrepetitive colorings of graphs and consider the connections
between this notion and the above results
ACMS 18th Biennial Conference Proceedings
Association of Christians in the Mathematical Sciences 18th Biennial Conference Proceedings, June 1-4, 2011, Westmont College, Santa Barbara, CA
First-Order Model Checking on Generalisations of Pushdown Graphs
We study the first-order model checking problem on two generalisations of
pushdown graphs. The first class is the class of nested pushdown trees. The
other is the class of collapsible pushdown graphs. Our main results are the
following. First-order logic with reachability is uniformly decidable on nested
pushdown trees. Considering first-order logic without reachability, we prove
decidability in doubly exponential alternating time with linearly many
alternations. First-order logic with regular reachability predicates is
uniformly decidable on level 2 collapsible pushdown graphs. Moreover, nested
pushdown trees are first-order interpretable in collapsible pushdown graphs of
level 2. This interpretation can be extended to an interpretation of the class
of higher-order nested pushdown trees in the collapsible pushdown graph
hierarchy. We prove that the second level of this new hierarchy of nested trees
has decidable first-order model checking. Our decidability result for
collapsible pushdown graph relies on the fact that level 2 collapsible pushdown
graphs are uniform tree-automatic. Our last result concerns tree-automatic
structures in general. We prove that first-order logic extended by Ramsey
quantifiers is decidable on all tree-automatic structures.Comment: phd thesis, 255 page
On the maximum intersecting sets of the general semilinear group of degree
Let be a prime and . A subset is intersecting if any two semilinear
transformations in agree on some non-zero vector in
. We show that any intersecting set of is of size at most that of a stabilizer of a non-zero vector, and we
characterize the intersecting sets of this size. Our proof relies on finding a
subgraph which is a lexicographic product in the derangement graph of
in its action on non-zero vectors of
. This method is also applied to give a new proof that the only
maximal intersecting sets of are the maximum
intersecting sets
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