10 research outputs found
An extension of Tur\'an's Theorem, uniqueness and stability
We determine the maximum number of edges of an -vertex graph with the
property that none of its -cliques intersects a fixed set .
For , the -partite Turan graph turns out to be the unique
extremal graph. For , there is a whole family of extremal graphs,
which we describe explicitly. In addition we provide corresponding stability
results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's
comments incorporate
Removing induced powers of cycles from a graph via fewest edits
What is the minimum proportion of edges which must be added to or removed
from a graph of density to eliminate all induced cycles of length ? The
maximum of this quantity over all graphs of density is measured by the edit
distance function, , a function which provides
a natural metric between graphs and hereditary properties.
Martin determined for all
when and determined
for . Peck determined
for all for odd cycles, and for
for even cycles. In this paper, we fully
determine the edit distance function for and . Furthermore, we
improve on the result of Peck for even cycles, by determining
for all ,
where for a constant . More generally, if is the
-th power of the cycle , we determine
for all in the case when
, thus improving on earlier work of Berikkyzy, Martin and Peck.Comment: 17 page
The edit distance function: Forbidding induced powers of cycles and other questions
The edit distance between two graphs on the same labeled vertex set is defined to be the size of the symmetric difference of the edge sets. The edit distance between a graph, , and a graph property, , is the minimum edit distance between and a graph in . The edit distance function of a graph property is a function of that measures, in the limit, the maximum normalized edit distance between a graph of density and .
In this thesis, we address the edit distance function for the property of having no induced copy of , the t^{\mbox{th}} power of the cycle of length . For and not divisible by , we determine the function for all values of . For and divisible by , the function is obtained for all but small values of . We also obtain some results for smaller values of , present alternative proofs of some important previous results using simple optimization techniques and discuss possible extension of the theory to hypergraphs
On the edit distance from a cycle- and squared cycle-free graph
The edit distance from a hereditary property is the fraction of edges in a graph that must be added or deleted for a graph to become a member of that hereditary property. Let Forb(Ch) and Forb(C2h) denote the hereditary properties containing graphs with no induced cycle or squared cycle on h vertices, respectively. The edit distance from Forb(Ch) is found for odd values of h, and the maximum edit distance is found for all values of h. The edit distance is found for Forb(C2h) for h = 8; 9; 10, and the maximum value is known for h = 11; 12, with partial results for other values of h
Spanning and induced subgraphs in graphs and digraphs
In this thesis, we make progress on three problems in extremal combinatorics, particularly in relation to finding large spanning subgraphs, and removing induced subgraphs.
First, we prove a generalisation of a result of Komlós, Sárközy and Szemerédi and show that for sufficiently large, any -vertex digraph with minimum semidegree at least contains a copy of every -vertex oriented tree with underlying maximum degree at most .
For our second result, we prove that when is an even integer and is sufficiently large, if is a -partite graph with vertex classes each of size and , then contains a transversal -factor, that is, a -factor in which each copy of contains exactly one vertex from each vertex class. In the case when is odd, we reduce the problem to proving that when is close to a specific extremal structure, it contains a transversal -factor. This resolves a conjecture of Fischer for even .
Our third result falls into the theory of edit distances. Let be the -th power of a cycle of length , that is, a cycle of length with additional edges between vertices at distance at most on the cycle. Let be the class of graphs with no induced copy of . For , what is the minimum proportion of edges which must be added to or removed from a graph of density to eliminate all induced copies of ? The maximum of this quantity over all graphs of density is measured by the edit distance function, , a function which provides a natural metric between graphs and hereditary properties. For our third result, we determine for all in the case when , where , thus improving on earlier work of Berikkyzy, Martin and Peck