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- All Subjects: Partially ordered sets
- All Subjects: Theoretical mathematics
- Creators: Czygrinow, Andrzej
Description
A central concept of combinatorics is partitioning structures with given constraints. Partitions of on-line posets and on-line graphs, which are dynamic versions of the more familiar static structures posets and graphs, are examined. In the on-line setting, vertices are continually added to a poset or graph while a chain partition or coloring (respectively) is maintained. %The optima of the static cases cannot be achieved in the on-line setting. Both upper and lower bounds for the optimum of the number of chains needed to partition a width $w$ on-line poset exist. Kierstead's upper bound of $\frac{5^w-1}{4}$ was improved to $w^{14 \lg w}$ by Bosek and Krawczyk. This is improved to $w^{3+6.5 \lg w}$ by employing the First-Fit algorithm on a family of restricted posets (expanding on the work of Bosek and Krawczyk) . Namely, the family of ladder-free posets where the $m$-ladder is the transitive closure of the union of two incomparable chains $x_1\le\dots\le x_m$, $y_1\le\dots\le y_m$ and the set of comparabilities $\{x_1\le y_1,\dots, x_m\le y_m\}$. No upper bound on the number of colors needed to color a general on-line graph exists. To lay this fact plain, the performance of on-line coloring of trees is shown to be particularly problematic. There are trees that require $n$ colors to color on-line for any positive integer $n$. Furthermore, there are trees that usually require many colors to color on-line even if they are presented without any particular strategy. For restricted families of graphs, upper and lower bounds for the optimum number of colors needed to maintain an on-line coloring exist. In particular, circular arc graphs can be colored on-line using less than 8 times the optimum number from the static case. This follows from the work of Pemmaraju, Raman, and Varadarajan in on-line coloring of interval graphs.
ContributorsSmith, Matthew Earl (Author) / Kierstead, Henry A (Thesis advisor) / Colbourn, Charles (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Hurlbert, Glenn (Committee member) / Arizona State University (Publisher)
Created2012
Description
Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.
In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.
Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.
ContributorsYie, Jangwon (Author) / Czygrinow, Andrzej (Thesis advisor) / Kierstead, Henry (Committee member) / Colbourn, Charles (Committee member) / Fishel, Susanna (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2018
Description
The Tamari lattices have been intensely studied since they first appeared in Dov Tamari’s thesis around 1952. He defined the n-th Tamari lattice T(n) on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Despite their interesting aspects and the attention they have received, a formula for the number of maximal chains in the Tamari lattices is still unknown. The purpose of this thesis is to convey my results on progress toward the solution of this problem and to discuss future work.
A few years ago, Bergeron and Préville-Ratelle generalized the Tamari lattices to the m-Tamari lattices. The original Tamari lattices T(n) are the case m=1. I establish a bijection between maximum length chains in the m-Tamari lattices and standard m-shifted Young tableaux. Using Thrall’s formula, I thus derive the formula for the number of maximum length chains in T(n).
For each i greater or equal to -1 and for all n greater or equal to 1, I define C(i,n) to be the set of maximal chains of length n+i in T(n). I establish several properties of maximal chains (treated as tableaux) and identify a particularly special property: each maximal chain may or may not possess a plus-full-set. I show, surprisingly, that for all n greater or equal to 2i+4, each member of C(i,n) contains a plus-full-set. Utilizing this fact and a collection of maps, I obtain a recursion for the number of elements in C(i,n) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3i+3. For example, the number of maximal chains of length n in T(n) is n choose 3.
I discuss current work and future plans involving certain equivalence classes of maximal chains in the Tamari lattices. If a maximal chain may be obtained from another by swapping a pair of consecutive edges with another pair in the Hasse diagram, the two maximal chains are said to differ by a square move. Two maximal chains are said to be in the same equivalence class if one may be obtained from the other by making a set of square moves.
A few years ago, Bergeron and Préville-Ratelle generalized the Tamari lattices to the m-Tamari lattices. The original Tamari lattices T(n) are the case m=1. I establish a bijection between maximum length chains in the m-Tamari lattices and standard m-shifted Young tableaux. Using Thrall’s formula, I thus derive the formula for the number of maximum length chains in T(n).
For each i greater or equal to -1 and for all n greater or equal to 1, I define C(i,n) to be the set of maximal chains of length n+i in T(n). I establish several properties of maximal chains (treated as tableaux) and identify a particularly special property: each maximal chain may or may not possess a plus-full-set. I show, surprisingly, that for all n greater or equal to 2i+4, each member of C(i,n) contains a plus-full-set. Utilizing this fact and a collection of maps, I obtain a recursion for the number of elements in C(i,n) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3i+3. For example, the number of maximal chains of length n in T(n) is n choose 3.
I discuss current work and future plans involving certain equivalence classes of maximal chains in the Tamari lattices. If a maximal chain may be obtained from another by swapping a pair of consecutive edges with another pair in the Hasse diagram, the two maximal chains are said to differ by a square move. Two maximal chains are said to be in the same equivalence class if one may be obtained from the other by making a set of square moves.
ContributorsNelson, Luke (Author) / Fishel, Susanna (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Jones, John (Committee member) / Kierstead, Henry (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2016
Description
The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum
number of colors needed to color $V(G)$ such that no adjacent vertices
receive the same color. The coloring number $\col(G)$ of a graph
$G$ is the minimum number $k$ such that there exists a linear ordering
of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.
It is well known that the coloring number is an upper bound for the
chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is
a generalization of the coloring number, and it was first introduced
by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$
is the minimum integer $k$ such that for some linear ordering $L$
of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller
vertices $u$ (with respect to $L$) with a path of length at most
$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.
The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$
is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$
if and only if the distance between $x$ and $y$ in $G$ is $3$.
This dissertation improves the best known upper bound of the
chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$
of planar graphs $G$, which is $105$, to $95$. It also improves
the best known lower bound, which is $7$, to $9$.
A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.
number of colors needed to color $V(G)$ such that no adjacent vertices
receive the same color. The coloring number $\col(G)$ of a graph
$G$ is the minimum number $k$ such that there exists a linear ordering
of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.
It is well known that the coloring number is an upper bound for the
chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is
a generalization of the coloring number, and it was first introduced
by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$
is the minimum integer $k$ such that for some linear ordering $L$
of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller
vertices $u$ (with respect to $L$) with a path of length at most
$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.
The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$
is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$
if and only if the distance between $x$ and $y$ in $G$ is $3$.
This dissertation improves the best known upper bound of the
chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$
of planar graphs $G$, which is $105$, to $95$. It also improves
the best known lower bound, which is $7$, to $9$.
A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.
ContributorsAlmulhim, Ahlam (Author) / Kierstead, Henry (Thesis advisor) / Sen, Arunabha (Committee member) / Richa, Andrea (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2020