8 research outputs found
Supersaturation Problem for Color-Critical Graphs
The \emph{Tur\'an function} \ex(n,F) of a graph is the maximum number
of edges in an -free graph with vertices. The classical results of
Tur\'an and Rademacher from 1941 led to the study of supersaturated graphs
where the key question is to determine , the minimum number of copies
of that a graph with vertices and \ex(n,F)+q edges can have.
We determine asymptotically when is \emph{color-critical}
(that is, contains an edge whose deletion reduces its chromatic number) and
.
Determining the exact value of seems rather difficult. For
example, let be the limit superior of for which the extremal
structures are obtained by adding some edges to a maximum -free graph.
The problem of determining for cliques was a well-known question of Erd\H
os that was solved only decades later by Lov\'asz and Simonovits. Here we prove
that for every {color-critical}~. Our approach also allows us to
determine for a number of graphs, including odd cycles, cliques with one
edge removed, and complete bipartite graphs plus an edge.Comment: 27 pages, 2 figure
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
A new bound for the 2/3 conjecture
We show that any n-vertex complete graph with edges colored with three colors
contains a set of at most four vertices such that the number of the neighbors
of these vertices in one of the colors is at least 2n/3. The previous best
value, proved by Erdos, Faudree, Gould, Gy\'arf\'as, Rousseau and Schelp in
1989, is 22. It is conjectured that three vertices suffice
On the Maximum Number of Edges in a Hypergraph with a Unique Perfect Matching
In this note, we determine the maximum number of edges of a k-uniform hypergraph, k\u3c3, with a unique perfect matching. This settles a conjecture proposed by Snevily
On the Maximum Number of Edges in a Hypergraph with a Unique Perfect Matching
In this note, we determine the maximum number of edges of a k-uniform hypergraph, k\u3c3, with a unique perfect matching. This settles a conjecture proposed by Snevily
Eccentricity of networks with structural constraints
The eccentricity of a node v in a network is the maximum distance from v to any other node. In social networks, the reciprocal of eccentricity is used as a measure of the importance of a node within a network. The associated centralization measure then calculates the degree to which a network is dominated by a particular node. In this work, we determine the maximum value of eccentricity centralization as well as the most centralized networks for various classes of networks including the families of bipartite networks (two-mode data) with given partition sizes and tree networks with fixed number of nodes and fixed maximum degree. To this end, we introduce and study a new way of enumerating the nodes of a tree which might be of independent interest
Eccentricity of networks with structural constraints
Fannie Simmons - wifehttps://stars.library.ucf.edu/cfm-ch-register-vol18/1609/thumbnail.jp