20 research outputs found
On List Coloring with Separation of the Complete Graph and Set System Intersections
We consider the following list coloring with separation problem: Given a
graph and integers , find the largest integer such that for any
list assignment of with for any vertex and for any edge of , there exists an assignment of
sets of integers to the vertices of such that and
for any vertex and
for any edge . Such a value of is called the separation number of
. Using a special partition of a set of lists for which we obtain an
improved version of Poincar\'e's crible, we determine the separation number of
the complete graph for some values of and , and prove bounds for
the remaining values.Comment: 18 page
Algorithmic approaches to problems in probabilistic combinatorics
The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method:
1. We study the structural properties of the triangle-free graphs generated by a semi-random variant of triangle-free process and obtain a packing extension of Kim's famous R(3,t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse.
2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n).
3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory.
4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.Ph.D
Coloring and covering problems on graphs
The \emph{separation dimension} of a graph , written , is the minimum number of linear orderings of such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} , which is the minimum of such that some linear orderings (repetition allowed) separate every two nonincident edges at least times.
In contrast to separation dimension, we show fractional separation dimension is bounded: always , with equality if and only if contains . There is no stronger bound even for bipartite graphs, since . We also compute for cycles and some complete tripartite graphs. We show that when is a tree and present a sequence of trees on which the value tends to . We conjecture that when the -free -vertex graph maximizing is .
We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let be the number of circular orderings needed to separate all pairs, and let be the fractional version. Among our results: (1) if and only is outerplanar. (2) when is bipartite. (3) . (4) , with equality if and only if . (5) .
A \emph{star -coloring} is a proper -coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring.
A proper vertex coloring of a graph is \emph{-dynamic} if for each , at least colors appear in . We investigate -dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs.
Given a proper total -coloring of a graph , we define the \emph{sum value} of a vertex to be . The smallest integer such that has a proper total -coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} . Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that for any simple graph with maximum degree . We prove this bound to be asymptotically correct by showing that . The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring