117 research outputs found
The homotopy type of the independence complex of graphs with no induced cycle of length divisible by
We prove Engstr\"{o}m's conjecture that the independence complex of graphs
with no induced cycle of length divisible by is either contractible or
homotopy equivalent to a sphere. Our result strengthens a result by Zhang and
Wu, verifying a conjecture of Kalai and Meshulam which states that the total
Betti number of the independence complex of such a graph is at most . A
weaker conjecture was proved earlier by Chudnovsky, Scott, Seymour, and Spikl,
who showed that in such a graph, the number of independent sets of even size
minus the number of independent sets of odd size has values , , or .Comment: 8 page
A system of disjoint representatives of line segments with given directions
We prove that for all positive integers and , there exists an integer
satisfying the following. If is a set of direction vectors
in the plane and is the set of all line segments in direction
for some , then for every families , each consisting of mutually disjoint segments in
, there is a set of disjoint segments
in and distinct integers satisfying that for all
. We generalize this property for underlying lines on
fixed directions to families of simple curves with certain conditions
Rainbow independent sets on dense graph classes
Given a family of independent sets in a graph, a rainbow
independent set is an independent set such that there is an injection
where for each , is contained in
. Aharoni, Briggs, J. Kim, and M. Kim [Rainbow independent sets in
certain classes of graphs. arXiv:1909.13143] determined for various graph
classes whether satisfies a property that for every
, there exists such that every family of
independent sets of size in a graph in contains a rainbow
independent set of size . In this paper, we add two dense graph classes
satisfying this property, namely, the class of graphs of bounded neighborhood
diversity and the class of -powers of graphs in a bounded expansion class
Badges and rainbow matchings
Drisko proved that matchings of size in a bipartite graph have a
rainbow matching of size . For general graphs it is conjectured that
matchings suffice for this purpose (and that matchings suffice when
is even). The known graphs showing sharpness of this conjecture for even
are called badges. We improve the previously best known bound from to
, using a new line of proof that involves analysis of the appearance of
badges. We also prove a "cooperative" generalization: for and ,
any sets of edges, the union of every of which contains a matching
of size , have a rainbow matching of size .Comment: Accepted for publication in Discrete Mathematics. 19 pages, 2 figure
Complementary Domain Adaptation and Generalization for Unsupervised Continual Domain Shift Learning
Continual domain shift poses a significant challenge in real-world
applications, particularly in situations where labeled data is not available
for new domains. The challenge of acquiring knowledge in this problem setting
is referred to as unsupervised continual domain shift learning. Existing
methods for domain adaptation and generalization have limitations in addressing
this issue, as they focus either on adapting to a specific domain or
generalizing to unseen domains, but not both. In this paper, we propose
Complementary Domain Adaptation and Generalization (CoDAG), a simple yet
effective learning framework that combines domain adaptation and generalization
in a complementary manner to achieve three major goals of unsupervised
continual domain shift learning: adapting to a current domain, generalizing to
unseen domains, and preventing forgetting of previously seen domains. Our
approach is model-agnostic, meaning that it is compatible with any existing
domain adaptation and generalization algorithms. We evaluate CoDAG on several
benchmark datasets and demonstrate that our model outperforms state-of-the-art
models in all datasets and evaluation metrics, highlighting its effectiveness
and robustness in handling unsupervised continual domain shift learning
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