An on-line competitive algorithm for coloring bipartite graphs without long induced paths

The existence of an on-line competitive algorithm for coloring bipartite graphs remains a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose a new on-line competitive coloring algorithm for $P_9$-free bipartite graphs

Planar posets have dimension at most linear in their height

We prove that every planar poset $P$ of height $h$ has dimension at most $192h + 96$. This improves on previous exponential bounds and is best possible up to a constant factor. We complement this result with a construction of planar posets of height $h$ and dimension at least $(4/3)h-2$.Comment: v2: Minor change

A note on concurrent graph sharing games

In the concurrent graph sharing game, two players, called First and Second, share the vertices of a connected graph with positive vertex-weights summing up to $1$ as follows. The game begins with First taking any vertex. In each proceeding round, the player with the smaller sum of collected weights so far chooses a non-taken vertex adjacent to a vertex which has been taken, i.e., the set of all taken vertices remains connected and one new vertex is taken in every round. (It is assumed that no two subsets of vertices have the same sum of weights.) One can imagine the players consume their taken vertex over a time proportional to its weight, before choosing a next vertex. In this note we show that First has a strategy to guarantee vertices of weight at least $1/3$ regardless of the graph and how it is weighted. This is best-possible already when the graph is a cycle. Moreover, if the graph is a tree First can guarantee vertices of weight at least $1/2$, which is clearly best-possible.Comment: expanded introduction and conclusion

Nowhere Dense Graph Classes and Dimension

Nowhere dense graph classes provide one of the least restrictive notions of sparsity for graphs. Several equivalent characterizations of nowhere dense classes have been obtained over the years, using a wide range of combinatorial objects. In this paper we establish a new characterization of nowhere dense classes, in terms of poset dimension: A monotone graph class is nowhere dense if and only if for every $h \geq 1$ and every $\epsilon > 0$, posets of height at most $h$ with $n$ elements and whose cover graphs are in the class have dimension $\mathcal{O}(n^{\epsilon})$.Comment: v4: Minor changes suggested by a refere

Burling graphs, chromatic number, and orthogonal tree-decompositions

A classic result of Asplund and Gr\"unbaum states that intersection graphs of axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmovi\'c, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note we provide a negative answer: There are graphs with arbitrarily large chromatic number for which one can find two tree-decompositions such that each bag of the first decomposition intersects each bag of the second in at most two vertices. Furthermore, this remains true even if one of the two decompositions is restricted to be a path-decomposition. This is shown using a construction of triangle-free graphs with unbounded chromatic number due to Burling, which we believe should be more widely known.Comment: v3: minor changes made following comments by the referees, v2: minor edit

On the dimension of posets with cover graphs of treewidth 22

In 1977, Trotter and Moore proved that a poset has dimension at most $3$ whenever its cover graph is a forest, or equivalently, has treewidth at most $1$. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth $3$. In this paper we focus on the boundary case of treewidth $2$. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth $2$ (Bir\'o, Keller, and Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover graph has treewidth $2$. We show that it is indeed the case: Every such poset has dimension at most $1276$.Comment: v4: minor changes made following helpful comments by the referee

Covergraphen und Ordnungsdimension

The goal of this dissertation is to study the various connections between the dimension of posets and graph theoretic properties of their cover graphs. We are particularly interested in properties of cover graphs that impose upper bounds on the dimension. A classical result in this context is due to Moore and Trotter from 1977. They showed that posets whose cover graphs are trees have dimension at most 3. Requiring cover graphs to be planar though is not enough to guarantee an upper bound on the dimension, as is witnessed by the famous construction of Kelly. However, Streib and Trotter proved that such a bound exists once the height of these posets is bounded. Their result forms the starting point of a series of papers and already led to a new fruitful research direction. In this dissertation we continue this line of research and determine the exact type of sparsity needed in cover graphs to guarantee that the dimension of the corresponding posets is bounded from above by a function in their height. This type of sparsity is given by the notion of bounded expansion, which was introduced by Nešetřil and Ossona de Mendez as a model for uniform sparseness in graphs. Our theorem generalizes a number of results including the most recent one for posets of bounded height with cover graphs excluding a fixed graph as a topological minor (Walczak, SODA 2015). We also show that the result is in a sense best possible, as it does not extend to nowhere dense classes; in fact, it already fails for cover graphs with locally bounded tree-width. The second main objective of this dissertation is to establish explicit bounds on the dimension in the cases that cover graphs satisfy concrete sparsity properties, such as being planar, having bounded tree-width, or excluding a fixed graph as a minor. Our main contribution in this context is an upper bound on the dimension of planar posets that is linear in height. This is an improvement upon exponential bounds and is best possible up to a constant factor as witnessed by Kelly's examples. We complement our results by several lower bound constructions certifying that our upper bounds are essentially best possible.In dieser Dissertation beschäftigen wir uns mit der Dimension von Partialordnungen und den zahlreichen Verbindungen dieses Parameters zu strukturellen Eigenschaften von Covergraphen. Ein klassisches Resultat in diesem Kontext stammt von Moore und Trotter aus dem Jahr 1977. Sie zeigen, dass Partialordnungen, deren Covergraphen Bäume sind, höchstens Dimension drei besitzen. Es ist bekannt, dass eine solche konstante obere Schranke an die Dimension nicht existiert, wenn wir jediglich fordern, dass die Covergraphen planar sind: Kellys Beispiele von planaren Partialordnungen aus dem Jahr 1981 haben beliebig große Dimension. Die Situation ändert sich, wenn wir die Höhe der Partialordnungen beschränken. Streib und Trotter zeigen, dass in diesem Fall die Dimension der Ordnungen nach oben beschränkt ist. Ihr Resultat begründete eine neue Forschingsrichtung in der Dimensionstheorie und steht mittlerweile am Anfang einer ganzen Reihe von Ergebnissen. In dieser Dissertation knüpfen wir an diese Ergebnisse an und bestimmen exakte strukturelle Eigenschaften an Covergraphen, die uns obere Schranken an die Dimension von zugehörigen Partialordnungen beschränkter Höhe liefern. Dabei folgen wir einer Hierarchie struktureller Eigenschaften auf Graphen, die auf Nešetřil und Ossona de Mendez zurückgeht. Wir zeigen, dass in dieser Hierarchie die Graphenklassen mit bounded expansion zuvor genannte Schranken garantieren und dass dies nicht mehr Fall ist, wenn Covergraphen nowhere dense sind. Beispiele von Graphenklassen mit bounded expansion sind gegeben durch planare Graphen, Graphen mit beschränkter Baumweite, und allgemeiner Graphen die einen fixen Graphen als topologischen Minor ausschließen. Sie gehen über diese jedoch hinaus, und somit verallgemeinert unser Theorem ein Reihe von Resultaten der letzten Jahre in diesem Bereich. Ein weiterer Schwerpunkt dieser Arbeit beschäftigt sich mit konkreten Abschätzungen an die Dimension von Partialordungen, deren Covergraphen aus den zuvor genannten Graphenklassen stammen. Unser Hauptresultat in dieser Richtung ist eine obere Schranke an die Dimension von planaren Partialordnungen, die linear in der Höhe der Ordnungen ist. Dieses Resultat ist bis auf einen konstanten Faktor bestmöglich, wie Kellys Beispiele zeigen, und verbessert somit bisherige exponentielle Schranken

Boxicity, poset dimension, and excluded minors

6 pages, 2 figuresInternational audienceIn this short note, we relate the boxicity of graphs (and the dimension of posets) with their generalized coloring parameters. In particular, together with known estimates, our results imply that any graph with no $K_t$-minor can be represented as the intersection of $O(t^2\log t)$ interval graphs (improving the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$ circular-arc graphs