Towards on-line Ohba's conjecture

The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs $G$ with $|V(G)| = 2 \chi(G)+1$ whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph $G$ with $|V(G)| \le 2 \chi(G)+1$ has its choice number equal its chromatic number. Nevertheless, an on-line version of Ohba conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: Every graph $G$ with $|V(G)| \le 2 \chi(G)$ has its on-line choice number equal its chromatic number. This paper confirms the on-line version of Ohba conjecture for graphs $G$ with independence number at most 3. We also study list colouring of complete multipartite graphs $K_{3\star k}$ with all parts of size 3. We prove that the on-line choice number of $K_{3 \star k}$ is at most $3/2k$, and present an alternate proof of Kierstead's result that its choice number is $\lceil (4k-1)/3 \rceil$. For general graphs $G$, we prove that if $|V(G)| \le \chi(G)+\sqrt{\chi(G)}$ then its on-line choice number equals chromatic number.Comment: new abstract and introductio

Chip games and paintability

We prove that the difference between the paint number and the choice number of a complete bipartite graph $K_{N,N}$ is $\Theta(\log \log N )$. That answers the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a k-uniform hypergraph with $\Theta(2^k )$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games

Pathwidth and nonrepetitive list coloring

A vertex coloring of a graph is nonrepetitive if there is no path in the graph whose first half receives the same sequence of colors as the second half. While every tree can be nonrepetitively colored with a bounded number of colors (4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently showed that this does not extend to the list version of the problem, that is, for every $\ell \geq 1$ there is a tree that is not nonrepetitively $\ell$-choosable. In this paper we prove the following positive result, which complements the result of Fiorenzi et al.: There exists a function $f$ such that every tree of pathwidth $k$ is nonrepetitively $f(k)$-choosable. We also show that such a property is specific to trees by constructing a family of pathwidth-2 graphs that are not nonrepetitively $\ell$-choosable for any fixed $\ell$.Comment: v2: Minor changes made following helpful comments by the referee

Deferred on-line bipartite matching

We present a new model for the problem of on-line matching on bipartite graphs. Suppose that one part of a graph is given, but the vertices of the other part are presented in an on-line fashion. In the classical version, each incoming vertex is either irrevocably matched to a vertex from the other part or stays unmatched forever. In our version, an algorithm is allowed to match the new vertex to a group of elements (possibly empty). Later on, the algorithm can decide to remove some vertices from the group and assign them to another (just presented) vertex, with the restriction that each element belongs to at most one group. We present an optimal (deterministic) algorithm for this problem and prove that its competitive ratio equals

A lazy approach to on-line bipartite matching

We present a new approach, called a lazy matching, to the problem of on-line matching on bipartite graphs. Imagine that one side of a graph is given and the vertices of the other side are arriving on-line. Originally, incoming vertex is either irrevocably matched to an another element or stays forever unmatched. A lazy algorithm is allowed to match a new vertex to a group of elements (possibly empty) and afterwords, forced against next vertices, may give up parts of the group. The restriction is that all the time each element is in at most one group. We present an optimal lazy algorithm (deterministic) and prove that its competitive ratio equals $1-\pi/\cosh(\frac{\sqrt{3}}{2}\pi)\approx 0.588$. The lazy approach allows us to break the barrier of $1/2$, which is the best competitive ratio that can be guaranteed by any deterministic algorithm in the classical on-line matching

Local computation algorithms for hypergraph coloring – following Beck’s approach

We investigate local computation algorithms (LCA) for two-coloring of k-uniform hypergraphs. We focus on hypergraph instances that satisfy strengthened assumption of the Lovász Local Lemma of the form $21−αk (∆ + 1)e < 1$, where ∆ is the bound on the maximum edge degree. The main question which arises here is for how large α there exists an LCA that is able to properly color such hypergraphs in polylogarithmic time per query. We describe briefly how upgrading the classical sequential procedure of Beck from 1991 with Moser and Tardos’ Resample yields polylogarithmic LCA that works for α up to 1/4. Then, we present an improved procedure that solves wider range of instances by allowing α up to 1/3

A Note on Two-Colorability of Nonuniform Hypergraphs

For a hypergraph $H$, let $q(H)$ denote the expected number of monochromatic edges when the color of each vertex in $H$ is sampled uniformly at random from the set of size 2. Let $s_{\min}(H)$ denote the minimum size of an edge in $H$. Erd\H{o}s asked in 1963 whether there exists an unbounded function $g(k)$ such that any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq g(k)$ is two colorable. Beck in 1978 answered this question in the affirmative for a function $g(k) = \Theta(\log^* k)$. We improve this result by showing that, for an absolute constant $\delta>0$, a version of random greedy coloring procedure is likely to find a proper two coloring for any hypergraph $H$ with $s_{\min}(H) \geq k$ and $q(H) \leq \delta \cdot \log k$