6 research outputs found
Coloring and constructing (hyper)graphs with restrictions
We consider questions regarding the existence of graphs and hypergraphs with certain coloring properties and other structural properties.
In Chapter 2 we consider color-critical graphs that are nearly bipartite and have few edges. We prove a conjecture of Chen, Erdős, Gyárfás, and Schelp concerning the minimum number of edges in a “nearly bipartite” 4-critical graph.
In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at most 2(k−1), then it is colorable from lists of size k; we prove that this is sharp, even with an additional girth requirement. Using the same approach, we also provide a simple construction of graphs with arbitrarily large girth and chromatic number (first proved to exist by Erdős).
In Chapter 4 we consider list-coloring the family of kth power graphs. Kostochka and Woodall conjectured that graph squares are chromatic-choosable, as a strengthening of the Total List Coloring Conjecture. Kim and Park disproved this stronger conjecture, and Zhu asked whether graph kth powers are chromatic-choosable for any k. We show that this is not true: we construct families of graphs based on affine planes whose choice number exceeds their chromatic number by a logarithmic factor.
In Chapter 5 we consider the existence of uniform hypergraphs with prescribed degrees and codegrees. In Section 5.2, we show that a generalization of the graphic 2-switch is insufficient to connect realizations of a given degree sequence. In Section 5.3, we consider an operation on 3-graphs related to the octahedron that preserves codegrees; this leads to an inductive definition for 2-colorable triangulations of the sphere. In Section 5.4, we discuss the notion of fractional realizations of degree sequences, in particular noting the equivalence of the existence of a realization and the existence of a fractional realization in the graph and multihypergraph cases.
In Chapter 6 we consider a question concerning poset dimension. Dorais asked for the maximum guaranteed size of a subposet with dimension at most d of an n-element poset. A lower bound of sqrt(dn) was observed by Goodwillie. We provide a sublinear upper bound
Generation and properties of random graphs and analysis of randomized algorithms
We study a new method of generating random -regular graphs by
repeatedly applying an operation called pegging. The pegging
algorithm, which applies the pegging operation in each step, is a
method of generating large random regular graphs beginning with
small ones. We prove that the limiting joint distribution of the
numbers of short cycles in the resulting graph is independent
Poisson. We use the coupling method to bound the total variation
distance between the joint distribution of short cycle counts and
its limit and thereby show that is an upper bound
of the \eps-mixing time. The coupling involves two different,
though quite similar, Markov chains that are not time-homogeneous.
We also show that the -mixing time is not
. This demonstrates that the upper bound
is essentially tight. We study also the
connectivity of random -regular graphs generated by the pegging
algorithm. We show that these graphs are asymptotically almost
surely -connected for any even constant .
The problem of orientation of random hypergraphs is motivated by the
classical load balancing problem. Let be two fixed integers.
Let \orH be a hypergraph whose hyperedges are uniformly of size
.
To {\em -orient} a hyperedge, we assign exactly of its
vertices positive signs with respect to this hyperedge, and the rest
negative. A -orientation of \orH consists of a
-orientation of all hyperedges of \orH, such that each vertex
receives at most positive signs from its incident hyperedges.
When is large enough, we determine the threshold of the
existence of a -orientation of a random hypergraph. The
-orientation of hypergraphs is strongly related to a general
version of the off-line load balancing problem.
The other topic we discuss is computing the probability of induced
subgraphs in a random regular graph. Let and be a graph
on vertices. For any with , we compute the
probability that the subgraph of induced by
is . The result holds for any and is further
extended to , the probability space of
random graphs with given degree sequence . This result
provides a basic tool for studying properties, for instance the
existence or the counts, of certain types of induced subgraphs
Core Structures in Random Graphs and Hypergraphs
The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results.
In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald.
We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges