123 research outputs found
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs
Loebl, Koml\'os and S\'os conjectured that every -vertex graph with at
least vertices of degree at least contains each tree of order
as a subgraph. We give a sketch of a proof of the approximate version of
this conjecture for large values of .
For our proof, we use a structural decomposition which can be seen as an
analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs.
With this tool, each graph can be decomposed into four parts: a set of vertices
of huge degree, regular pairs (in the sense of the regularity lemma), and two
other objects each exhibiting certain expansion properties. We then exploit the
properties of each of the parts of to embed a given tree .
The purpose of this note is to highlight the key steps of our proof. Details
can be found in [arXiv:1211.3050]
Tight bounds towards a conjecture of Gallai
We prove that for , if is an -vertex graph with chromatic
number but any its proper subgraph has smaller chromatic number, then
contains at most copies of cliques of size . This answers a
problem of Abbott and Zhou and provides a tight bound on a conjecture of
Gallai
Hypohamiltonian and almost hypohamiltonian graphs
This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results
Replication in critical graphs and the persistence of monomial ideals
Motivated by questions about square-free monomial ideals in polynomial rings,
in 2010 Francisco et al. conjectured that for every positive integer k and
every k-critical (i.e., critically k-chromatic) graph, there is a set of
vertices whose replication produces a (k+1)-critical graph. (The replication of
a set W of vertices of a graph is the operation that adds a copy of each vertex
w in W, one at a time, and connects it to w and all its neighbours.)
We disprove the conjecture by providing an infinite family of
counterexamples. Furthermore, the smallest member of the family answers a
question of Herzog and Hibi concerning the depth functions of square-free
monomial ideals in polynomial rings, and a related question on the persistence
property of such ideals
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