6 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. Bollobás 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 Bollobás 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
Two stability theorems for -saturated hypergraphs
An -saturated -graph is a maximal -graph not containing
any member of as a subgraph. Let be
the collection of all -graphs with at most edges
such that for some -set every pair is covered by an edge in . Our first result shows that for each every -saturated -graph on
vertices with edges contains a complete
-partite subgraph on vertices, which extends a stability
theorem for -saturated graphs given by Popielarz, Sahasrabudhe and
Snyder. We also show that the bound is best possible. Our second result is
motivated by a celebrated theorem of Andr\'{a}sfai, Erd\H{o}s and S\'{o}s which
states that for every -free graph on vertices
with minimum degree is -partite.
We give a hypergraph version of it. The \emph{minimum positive co-degree} of an
-graph , denoted by , is the
maximum such that if is an -set contained in a edge of
, then is contained in at least distinct edges of
. Let be an integer and be a
-saturated -graph on vertices. We prove that if
either and ; or and , then is -partite; and the bound is best possible.
This is the first stability result on minimum positive co-degree for
hypergraphs