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

Molecular hydrogen isotopes adsorbed on krypton-preplated graphite: Quantum Monte Carlo simulations

Adsorption of ortho-deuterium and para-hydrogen films on a graphite substrate, pre-plated with a single atomic layer of krypton, is studied theoretically by means of quantum Monte Carlo simulations at low temperature. Our model explicitly includes substrate corrugation. Energetic and structural properties of these adsorbed films are computed for a range of hydrogen coverages. Thermodynamically stable adsorbed films are solid, with no clear evidence of any liquid-like phase. Quantum exchanges of ortho-deuterium and para-hydrogen are essentially absent in this system, down to zero temperature; consequently, this system displays no superfluidity in this limit. Our simulations provide evidence of a stable domain wall fluid at low temperature, consistently with recent experimental observations.Comment: 7 pages, 7 figure

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

Beneficial use of off-specification fly ashes to increase the shear strength and stiffness of expansive soil-rubber (ESR) mixtures

2011 Spring.Includes bibliographical references.The potential use of off-specification fly ashes to increase the shear strength and stiffness of an expansive soil-rubber (ESR) mixture was investigated systematically in this study. The off-specification fly ashes used included a high sulfur content fly ash and a high carbon content fly ash. A standard Class C fly ash was also used as a control fly ash to develop a basis for comparison of the effects of the off-specification fly ashes. The ESR mixture consisted of high-plasticity clay blended with 20% 6.7-mm granulated rubber (by weight). The fly ash content required to develop pozzolanic reactions was determined based on the concept of lime fixation point and kept constant for all ESR-fly ash mixtures. At this selected fly ash content, ESR-fly ash mixtures were tested at a single relative compaction level and curing times of 7 and 14 days. Unconfined compression testing was performed on compacted specimens to validate the fly ash content selected and the effect of curing time on the development of pozzolanic reactions. The effect of the fly ash type, curing time and mean effective stress was evaluated by performing isotropically consolidated undrained triaxial compression tests on saturated specimens at mean effective stress levels of 50, 100 and 200 kPa. Stiffness changes due to fly ash addition were evaluated during undrained compression. Large-strain stiffness was measured using conventional external displacement transducers. Very-small strain stiffness was evaluated from shear wave velocity measurements using a bender element apparatus. Results suggest that the shear strength and stiffness improvements imparted by the off-specification fly ashes is similar to or better than the improvements imparted by conventional Class C fly ash

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