213 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
Balanced supersaturation for some degenerate hypergraphs
A classical theorem of Simonovits from the 1980s asserts that every graph
satisfying must contain copies of . Recently, Morris and
Saxton established a balanced version of Simonovits' theorem, showing that such
has copies of , which
are `uniformly distributed' over the edges of . Moreover, they used this
result to obtain a sharp bound on the number of -free graphs via the
container method. In this paper, we generalise Morris-Saxton's results for even
cycles to -graphs. We also prove analogous results for complete
-partite -graphs.Comment: Changed title, abstract and introduction were rewritte
Extremal graphs for the odd prism
The Tur\'an number of a graph is the maximum number of
edges in an -vertex graph which does not contain as a subgraph. The
Tur\'{a}n number of regular polyhedrons was widely studied in a series of works
due to Simonovits. In this paper, we shall present the exact Tur\'{a}n number
of the prism , which is defined as the Cartesian product
of an odd cycle and an edge . Applying a deep theorem of
Simonovits and a stability result of Yuan [European J. Combin. 104 (2022)], we
shall determine the exact value of for
every and sufficiently large , and we also characterize the
extremal graphs. Moreover, in the case of , motivated by a recent result
of Xiao, Katona, Xiao and Zamora [Discrete Appl. Math. 307 (2022)], we will
determine the exact value of for every
instead of for sufficiently large .Comment: 24 page
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