8,356 research outputs found
Two new Ramsey numbers
A graph with many vertices cannot be homogeneous, i.e., for any pair of integers (i,j) all large graphs must contain either a complete subgraph on i vertices or an independent set of size j. The Ramsey number for (i,j) is the smallest integer R such that all graphs with at least R vertices have this property. For example, the (3,3) Ramsey number is 6; if a graph has 6 or more vertices, then is must contain a triangle or an independent set of size 3. The (4,4) Ramsey number is 18, found in 1954 [GG] . The (5,5) Ramsey number is still unknown; it is between 43 and 52. This thesis deals with subgraphs slightly different from the classical types. The subgraphs here are complete graphs with one edge missing and induced subgraphs with exactly one edge. The (4,6) and (4,7) Ramsey numbers for these types of subgraphs is computed. The method used is an exhaustive search, with many shortcuts employed to reduce computation time
Large cliques or cocliques in hypergraphs with forbidden order-size pairs
The well-known Erdős-Hajnal conjecture states that for any graph , there exists such that every -vertex graph that contains no induced copy of has a homogeneous set of size at least . We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on vertices and edges for any positive and , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case , for every , we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in . In most cases the bounds are essentially tight. We also determine, for all , whether the growth rate is polynomial or polylogarithmic. Some open problems remain
Large cliques or co-cliques in hypergraphs with forbidden order-size pairs
The well-known Erd\H{o}s-Hajnal conjecture states that for any graph ,
there exists such that every -vertex graph that contains no
induced copy of has a homogeneous set of size at least . We
consider a variant of the Erd\H{o}s-Hajnal problem for hypergraphs where we
forbid a family of hypergraphs described by their orders and sizes. For graphs,
we observe that if we forbid induced subgraphs on vertices and edges
for any positive and , then we obtain large
homogeneous sets. For triple systems, in the first nontrivial case , for
every , we give bounds on the minimum size of a
homogeneous set in a triple system where the number of edges spanned by every
four vertices is not in . In most cases the bounds are essentially tight. We
also determine, for all , whether the growth rate is polynomial or
polylogarithmic. Some open problems remain.Comment: A preliminary version of this manuscript appeared as arXiv:2303.0957
Unavoidable induced subgraphs in large graphs with no homogeneous sets
A homogeneous set of an -vertex graph is a set of vertices () such that every vertex not in is either complete or
anticomplete to . A graph is called prime if it has no homogeneous set. A
chain of length is a sequence of vertices such that for every vertex
in the sequence except the first one, its immediate predecessor is its unique
neighbor or its unique non-neighbor among all of its predecessors. We prove
that for all , there exists such that every prime graph with at least
vertices contains one of the following graphs or their complements as an
induced subgraph: (1) the graph obtained from by subdividing every
edge once, (2) the line graph of , (3) the line graph of the graph in
(1), (4) the half-graph of height , (5) a prime graph induced by a chain of
length , (6) two particular graphs obtained from the half-graph of height
by making one side a clique and adding one vertex.Comment: 13 pages, 3 figure
The random subgraph model for the analysis of an ecclesiastical network in Merovingian Gaul
In the last two decades many random graph models have been proposed to
extract knowledge from networks. Most of them look for communities or, more
generally, clusters of vertices with homogeneous connection profiles. While the
first models focused on networks with binary edges only, extensions now allow
to deal with valued networks. Recently, new models were also introduced in
order to characterize connection patterns in networks through mixed
memberships. This work was motivated by the need of analyzing a historical
network where a partition of the vertices is given and where edges are typed. A
known partition is seen as a decomposition of a network into subgraphs that we
propose to model using a stochastic model with unknown latent clusters. Each
subgraph has its own mixing vector and sees its vertices associated to the
clusters. The vertices then connect with a probability depending on the
subgraphs only, while the types of edges are assumed to be sampled from the
latent clusters. A variational Bayes expectation-maximization algorithm is
proposed for inference as well as a model selection criterion for the
estimation of the cluster number. Experiments are carried out on simulated data
to assess the approach. The proposed methodology is then applied to an
ecclesiastical network in Merovingian Gaul. An R code, called Rambo,
implementing the inference algorithm is available from the authors upon
request.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS691 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Gains in Power from Structured Two-Sample Tests of Means on Graphs
We consider multivariate two-sample tests of means, where the location shift
between the two populations is expected to be related to a known graph
structure. An important application of such tests is the detection of
differentially expressed genes between two patient populations, as shifts in
expression levels are expected to be coherent with the structure of graphs
reflecting gene properties such as biological process, molecular function,
regulation, or metabolism. For a fixed graph of interest, we demonstrate that
accounting for graph structure can yield more powerful tests under the
assumption of smooth distribution shift on the graph. We also investigate the
identification of non-homogeneous subgraphs of a given large graph, which poses
both computational and multiple testing problems. The relevance and benefits of
the proposed approach are illustrated on synthetic data and on breast cancer
gene expression data analyzed in context of KEGG pathways
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