An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains

It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w then Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound of ckw^{2} on the number of chains used by First-Fit for some constant c, while Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this paper we prove an upper bound of the form ckw. This is best possible up to the value of c.Comment: v3: referees' comments incorporate

An extremal problem on crossing vectors

For positive integers $w$ and $k$, two vectors $A$ and $B$ from $\mathbb{Z}^w$ are called $k$-crossing if there are two coordinates $i$ and $j$ such that $A[i]-B[i]\geq k$ and $B[j]-A[j]\geq k$. What is the maximum size of a family of pairwise $1$-crossing and pairwise non-$k$-crossing vectors in $\mathbb{Z}^w$? We state a conjecture that the answer is $k^{w-1}$. We prove the conjecture for $w\leq 3$ and provide weaker upper bounds for $w\geq 4$. Also, for all $k$ and $w$, we construct several quite different examples of families of desired size $k^{w-1}$. This research is motivated by a natural question concerning the width of the lattice of maximum antichains of a partially ordered set.Comment: Corrections and improvement

On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.Comment: 16 pages, 10 figure

On-line coloring between two lines

We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses $O(\omega^3)$ colors on graphs with maximum clique size $\omega$. In contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when $\omega=2$. The {\em left-of} relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class $\mathcal{C}$ of posets obtained from convex sets between two lines with some other classes of posets: all $2$-dimensional posets and all posets of height $2$ are in $\mathcal{C}$ but there is a $3$-dimensional poset of height $3$ that does not belong to $\mathcal{C}$. We also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.Comment: grant support adde

Topological minors of cover graphs and dimension

We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that $(k+k)$-free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of $k$.Comment: revised versio

Dimension of posets with planar cover graphs excluding two long incomparable chains

It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.Comment: New section on connections with graph minors, small correction