249 research outputs found
Bounds for the number of meeting edges in graph partitioning
summary:Let be a weighted hypergraph with edges of size at most 2. BollobΓ‘s and Scott conjectured that admits a bipartition such that each vertex class meets edges of total weight at least , where is the total weight of edges of size and is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph (i.e., multi-hypergraph), we show that there exists a bipartition of such that each vertex class meets edges of total weight at least , where is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with edges, except for and , admits a tripartition such that each vertex class meets at least edges, which establishes a special case of a more general conjecture of BollobΓ‘s and Scott
Counting triangles in graphs without vertex disjoint odd cycles
Given two graphs and , the maximum possible number of copies of in
an -free graph on vertices is denoted by . Let
denote vertex disjoint copies of . In this
paper, we determine the exact value of and its extremal graph, which generalizes some known results
MaxCut in graphs with sparse neighborhoods
Let be a graph with edges and let denote the size of
a largest cut of . The difference is called the surplus
of . A fundamental problem in MaxCut is to determine
for without specific structure, and the degree sequence
of plays a key role in getting the lower bound of
. A classical example, given by Shearer, is that
for triangle-free graphs ,
implying that . It was extended to graphs with
sparse neighborhoods by Alon, Krivelevich and Sudakov. In this paper, we
establish a novel and stronger result for a more general family of graphs with
sparse neighborhoods.
Our result can derive many well-known bounds on in -free
graphs for different , such as the triangle, the even cycle, the graphs
having a vertex whose removal makes the graph acyclic, or the complete
bipartite graph with . It can also deduce many new
(tight) bounds on in -free graphs when is any graph
having a vertex whose removal results in a bipartite graph with relatively
small Tur\'{a}n number, especially the even wheel. This contributes to a
conjecture raised by Alon, Krivelevich and Sudakov. Moreover, we give a new
family of graphs such that for
some constant in -free graphs , giving an evidence to a
conjecture suggested by Alon, Bollob\'as, Krivelevich and Sudakov
Hypergraphs with irrational Tur\'{a}n density and many extremal configurations
Unlike graphs, determining Tur\'{a}n densities of hypergraphs is known to be
notoriously hard in general. The essential reason is that for many classical
families of -uniform hypergraphs , there are perhaps many
near-extremal -free configurations with very different
structure. Such a phenomenon is called not stable, and Liu and Mubayi gave a
first not stable example. Another perhaps reason is that little is known about
the set consisting of all possible Tur\'{a}n densities which has cardinality of
the continuum. Let be an integer. In this paper, we construct a finite
family of 3-uniform hypergraphs such that the Tur\'{a}n density
of is irrational, and there are near-extremal
-free configurations that are far from each other in
edit-distance. This is the first not stable example that has an irrational
Tur\'{a}n density. It also provides a new phenomenon about feasible region
functions.Comment: arXiv admin note: text overlap with arXiv:2206.0394
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