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- All Subjects: Graph coloring
- Creators: Czygrinow, Andrzej
Description
Every graph can be colored with one more color than its maximum degree. A well-known theorem of Brooks gives the precise conditions under which a graph can be colored with maximum degree colors. It is natural to ask for the required conditions on a graph to color with one less color than the maximum degree; in 1977 Borodin and Kostochka conjectured a solution for graphs with maximum degree at least 9: as long as the graph doesn't contain a maximum-degree-sized clique, it can be colored with one fewer than the maximum degree colors. This study attacks the conjecture on multiple fronts. The first technique is an extension of a vertex shuffling procedure of Catlin and is used to prove the conjecture for graphs with edgeless high vertex subgraphs. This general approach also bears more theoretical fruit. The second technique is an extension of a method Kostochka used to reduce the Borodin-Kostochka conjecture to the maximum degree 9 case. Results on the existence of independent transversals are used to find an independent set intersecting every maximum clique in a graph. The third technique uses list coloring results to exclude induced subgraphs in a counterexample to the conjecture. The classification of such excludable graphs that decompose as the join of two graphs is the backbone of many of the results presented here. The fourth technique uses the structure theorem for quasi-line graphs of Chudnovsky and Seymour in concert with the third technique to prove the Borodin-Kostochka conjecture for claw-free graphs. The fifth technique adds edges to proper induced subgraphs of a minimum counterexample to gain control over the colorings produced by minimality. The sixth technique adapts a recoloring technique originally developed for strong coloring by Haxell and by Aharoni, Berger and Ziv to general coloring. Using this recoloring technique, the Borodin-Kostochka conjectured is proved for graphs where every vertex is in a large clique. The final technique is naive probabilistic coloring as employed by Reed in the proof of the Borodin-Kostochka conjecture for large maximum degree. The technique is adapted to prove the Borodin-Kostochka conjecture for list coloring for large maximum degree.
ContributorsRabern, Landon (Author) / Kierstead, Henry (Thesis advisor) / Colbourn, Charles (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Hurlbert, Glenn (Committee member) / Arizona State University (Publisher)
Created2013
Description
Graph coloring is about allocating resources that can be shared except where there are certain pairwise conflicts between recipients. The simplest coloring algorithm that attempts to conserve resources is called first fit. Interval graphs are used in models for scheduling (in computer science and operations research) and in biochemistry for one-dimensional molecules such as genetic material. It is not known precisely how much waste in the worst case is due to the first-fit algorithm for coloring interval graphs. However, after decades of research the range is narrow. Kierstead proved that the performance ratio R is at most 40. Pemmaraju, Raman, and Varadarajan proved that R is at most 10. This can be improved to 8. Witsenhausen, and independently Chrobak and Slusarek, proved that R is at least 4. Slusarek improved this to 4.45. Kierstead and Trotter extended the method of Chrobak and Slusarek to one good for a lower bound of 4.99999 or so. The method relies on number sequences with a certain property of order. It is shown here that each sequence considered in the construction satisfies a linear recurrence; that R is at least 5; that the Fibonacci sequence is in some sense minimally useless for the construction; and that the Fibonacci sequence is a point of accumulation in some space for the useful sequences of the construction. Limitations of all earlier constructions are revealed.
ContributorsSmith, David A. (Author) / Kierstead, Henry A. (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Gelb, Anne (Committee member) / Hurlbert, Glenn H. (Committee member) / Kadell, Kevin W. J. (Committee member) / Arizona State University (Publisher)
Created2010
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
Description
Let $G=(V,E)$ be a graph. A \emph{list assignment} $L$ for $G$ is a function from
$V$ to subsets of the natural numbers. An $L$-\emph{coloring} is a function $f$
with domain $V$ such that $f(v)\in L(v)$ for all vertices $v\in V$ and $f(x)\ne f(y)$
whenever $xy\in E$. If $|L(v)|=t$ for all $v\in V$ then $L$ is a $t$-\emph{list
assignment}. The graph $G$ is $t$-choosable if for every $t$-list assignment $L$
there is an $L$-coloring. The least $t$ such that $G$ is $t$-choosable is called
the list chromatic number of $G$, and is denoted by $\ch(G)$. The complete multipartite
graph with $k$ parts, each of size $s$ is denoted by $K_{s*k}$. Erd\H{o}s et al.
suggested the problem of determining $\ensuremath{\ch(K_{s*k})}$, and showed that
$\ch(K_{2*k})=k$. Alon gave bounds of the form $\Theta(k\log s)$. Kierstead proved
the exact bound $\ch(K_{3*k})=\lceil\frac{4k-1}{3}\rceil$. Here it is proved that
$\ch(K_{4*k})=\lceil\frac{3k-1}{2}\rceil$.
An online version of the list coloring problem was introduced independently by Schauz
and Zhu. It can be formulated as a game between two players, Alice and Bob. Alice
designs lists of colors for all vertices, but does not tell Bob, and is allowed to
change her mind about unrevealed colors as the game progresses. On her $i$-th turn
Alice reveals all vertices with $i$ in their list. On his $i$-th turn Bob decides,
irrevocably, which (independent set) of these vertices to color with $i$. For a
function $l$ from $V$ to the natural numbers, Bob wins the $l$-\emph{game} if
eventually he colors every vertex $v$ before $v$ has had $l(v)+1$ colors of its
list revealed by Alice; otherwise Alice wins. The graph $G$ is $l$-\emph{online
choosable} or \emph{$l$-paintable} if Bob has a strategy to win the $l$-game. If
$l(v)=t$ for all $v\in V$ and $G$ is $l$-paintable, then $G$ is t-paintable.
The \emph{online list chromatic number }of $G$ is the least $t$ such that $G$
is $t$-paintable, and is denoted by $\ensuremath{\ch^{\mathrm{OL}}(G)}$. Evidently,
$\ch^{\mathrm{OL}}(G)\geq\ch(G)$. Zhu conjectured that the gap $\ch^{\mathrm{OL}}(G)-\ch(G)$
can be arbitrarily large. However there are only a few known examples with this gap
equal to one, and none with larger gap. This conjecture is explored in this thesis.
One of the obstacles is that there are not many graphs whose exact list coloring
number is known. This is one of the motivations for establishing new cases of Erd\H{o}s'
problem. Here new examples of graphs with gap one are found, and related technical
results are developed as tools for attacking Zhu's conjecture.
The square $G^{2}$ of a graph $G$ is formed by adding edges between all vertices
at distance $2$. It was conjectured that every graph $G$ satisfies $\chi(G^{2})=\ch(G^{2})$.
This was recently disproved for specially constructed graphs. Here it is shown that
a graph arising naturally in the theory of cellular networks is also a counterexample.
$V$ to subsets of the natural numbers. An $L$-\emph{coloring} is a function $f$
with domain $V$ such that $f(v)\in L(v)$ for all vertices $v\in V$ and $f(x)\ne f(y)$
whenever $xy\in E$. If $|L(v)|=t$ for all $v\in V$ then $L$ is a $t$-\emph{list
assignment}. The graph $G$ is $t$-choosable if for every $t$-list assignment $L$
there is an $L$-coloring. The least $t$ such that $G$ is $t$-choosable is called
the list chromatic number of $G$, and is denoted by $\ch(G)$. The complete multipartite
graph with $k$ parts, each of size $s$ is denoted by $K_{s*k}$. Erd\H{o}s et al.
suggested the problem of determining $\ensuremath{\ch(K_{s*k})}$, and showed that
$\ch(K_{2*k})=k$. Alon gave bounds of the form $\Theta(k\log s)$. Kierstead proved
the exact bound $\ch(K_{3*k})=\lceil\frac{4k-1}{3}\rceil$. Here it is proved that
$\ch(K_{4*k})=\lceil\frac{3k-1}{2}\rceil$.
An online version of the list coloring problem was introduced independently by Schauz
and Zhu. It can be formulated as a game between two players, Alice and Bob. Alice
designs lists of colors for all vertices, but does not tell Bob, and is allowed to
change her mind about unrevealed colors as the game progresses. On her $i$-th turn
Alice reveals all vertices with $i$ in their list. On his $i$-th turn Bob decides,
irrevocably, which (independent set) of these vertices to color with $i$. For a
function $l$ from $V$ to the natural numbers, Bob wins the $l$-\emph{game} if
eventually he colors every vertex $v$ before $v$ has had $l(v)+1$ colors of its
list revealed by Alice; otherwise Alice wins. The graph $G$ is $l$-\emph{online
choosable} or \emph{$l$-paintable} if Bob has a strategy to win the $l$-game. If
$l(v)=t$ for all $v\in V$ and $G$ is $l$-paintable, then $G$ is t-paintable.
The \emph{online list chromatic number }of $G$ is the least $t$ such that $G$
is $t$-paintable, and is denoted by $\ensuremath{\ch^{\mathrm{OL}}(G)}$. Evidently,
$\ch^{\mathrm{OL}}(G)\geq\ch(G)$. Zhu conjectured that the gap $\ch^{\mathrm{OL}}(G)-\ch(G)$
can be arbitrarily large. However there are only a few known examples with this gap
equal to one, and none with larger gap. This conjecture is explored in this thesis.
One of the obstacles is that there are not many graphs whose exact list coloring
number is known. This is one of the motivations for establishing new cases of Erd\H{o}s'
problem. Here new examples of graphs with gap one are found, and related technical
results are developed as tools for attacking Zhu's conjecture.
The square $G^{2}$ of a graph $G$ is formed by adding edges between all vertices
at distance $2$. It was conjectured that every graph $G$ satisfies $\chi(G^{2})=\ch(G^{2})$.
This was recently disproved for specially constructed graphs. Here it is shown that
a graph arising naturally in the theory of cellular networks is also a counterexample.
ContributorsWang, Ran (Author) / Kierstead, H.A. (Thesis advisor) / Colbourn, Charles (Committee member) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Sen, Arunabha (Committee member) / Arizona State University (Publisher)
Created2015