50 research outputs found
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
An updated survey on rainbow connections of graphs - a dynamic survey
The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow -connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study
Mark Sequences In Digraphs
In Chapter 1, we present a brief introduction of digraphs and some def-
initions. Chapter 2 is a review of scores in tournaments and oriented graphs.
Also we have obtained several new results on oriented graph scores and we
have given a new proof of Avery's theorem on oriented graph scores. In chap-
ter 3, we have introduced the concept of marks in multidigraphs, non-negative
integers attached to the vertices of multidigraphs. We have obtained several
necessary and su cient conditions for sequences of non-negative integers to
be mark sequences of some r-digraphs. We have derived stronger inequalities
for these marks. Further we have characterized uniquely mark sequences in
r-digraphs. This concept of marks has been extended to bipartite multidi-
graphs and multipartite multidigraphs in chapter 4. There we have obtained
characterizations for mark sequences in these types of multidigraphs and we
have given algorithms for constructing corresponding multidigraphs. Chap-
ter 5 deals with imbalances and imbalance sequences in digraphs. We have
generalized the concept of imbalances to oriented bipartite graphs and have
obtained criteria for a pair of integers to be the pair of imbalance sequences
of some oriented bipartite graph. We have shown the existence of an oriented
bipartite graph whose imbalance set is the given set of integers