403,689 research outputs found
Triangles in graphs without bipartite suspensions
Given graphs and , the generalized Tur\'an number ex is the
maximum number of copies of in an -vertex graph with no copies of .
Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of
ex when the chromatic number of is greater than 3 and proved
several results when is bipartite. We consider this problem when has
chromatic number 3. Even this special case for the following relatively simple
3-chromatic graphs appears to be challenging.
The suspension of a graph is the graph obtained from by
adding a new vertex adjacent to all vertices of . We give new upper and
lower bounds on ex when is a path, even cycle, or
complete bipartite graph. One of the main tools we use is the triangle removal
lemma, but it is unclear if much stronger statements can be proved without
using the removal lemma.Comment: New result about path with 5 edges adde
Graph-based task libraries for robots: generalization and autocompletion
In this paper, we consider an autonomous robot that persists
over time performing tasks and the problem of providing one additional
task to the robot's task library. We present an approach to generalize
tasks, represented as parameterized graphs with sequences, conditionals,
and looping constructs of sensing and actuation primitives. Our approach
performs graph-structure task generalization, while maintaining task ex-
ecutability and parameter value distributions. We present an algorithm
that, given the initial steps of a new task, proposes an autocompletion
based on a recognized past similar task. Our generalization and auto-
completion contributions are eective on dierent real robots. We show
concrete examples of the robot primitives and task graphs, as well as
results, with Baxter. In experiments with multiple tasks, we show a sig-
nicant reduction in the number of new task steps to be provided
A general theorem in spectral extremal graph theory
The extremal graphs and spectral extremal graphs
are the sets of graphs on vertices with
maximum number of edges and maximum spectral radius, respectively, with no
subgraph in . We prove a general theorem which allows us to
characterize the spectral extremal graphs for a wide range of forbidden
families and implies several new and existing results. In
particular, whenever contains the complete
bipartite graph (or certain similar graphs) then
contains the same graph when is sufficiently
large. We prove a similar theorem which relates
and , the set of -free graphs
which maximize the spectral radius of the matrix , where is the adjacency matrix and is the diagonal
degree matrix
Generalized Tur\'an problems for even cycles
Given a graph and a set of graphs , let
denote the maximum possible number of copies of in an -free
graph on vertices. We investigate the function , when
and members of are cycles. Let denote the cycle of
length and let . Some of our main
results are the following.
(i) We show that for any .
Moreover, we determine it asymptotically in the following cases: We show that
and that the maximum
possible number of 's in a -free bipartite graph is .
(ii) Solymosi and Wong proved that if Erd\H{o}s's Girth Conjecture holds,
then for any we have . We prove that forbidding any other even cycle
decreases the number of 's significantly: For any , we have
More generally,
we show that for any and such that , we have
(iii) We prove provided a
strong version of Erd\H{o}s's Girth Conjecture holds (which is known to be true
when ). Moreover, forbidding one more cycle decreases the number
of 's significantly: More precisely, we have and for .
(iv) We also study the maximum number of paths of given length in a
-free graph, and prove asymptotically sharp bounds in some cases.Comment: 37 Pages; Substantially revised, contains several new results.
Mistakes corrected based on the suggestions of a refere
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