arXiv subjects

Grzegorz Gutowski

Publications and source records attributed to Grzegorz Gutowski.

27 records · Page 2Linked to original sources

arXiv2018

The Slow-coloring Game on Sparse Graphs: $k$-Degenerate, Planar, and Outerplanar

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar.

math.CO↗

arXiv2018

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 $H$. Erdős 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 $g(k) = Θ(\log^* k)$. We improve this result by showing that, for an absolute constant $δ>0$, 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 $q(H) \leq δ\cdot \log k$.

math.CO↗

arXiv2017

Plane Graphs are Facially-non-repetitively $10^{4 \cdot10^7}$-Choosable

A sequence $\left(x_1,x_2,\ldots,x_{2n}\right)$ of even length is a repetition if $\left(x_1,\ldots,x_n\right) = \left(x_{n+1},\ldots,x_{2n}\right)$. We prove existence of a constant $C < 10^{4 \cdot 10^7}$ such that given any planar drawing of a graph $G$, and a list $L(v)$ of $C$ permissible colors for each vertex $v$ in $G$, there is a choice of a permissible color for each vertex such that the sequence of colors of the vertices on any facial simple path in $G$ is not a repetition.

math.CO↗

arXiv2017

Defective 3-Paintability of Planar Graphs

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.

math.CO↗

arXiv2016

Universal targets for homomorphisms of edge-colored graphs

A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class $\mathcal{F}$ of graphs, a $k$-edge-colored graph $\mathbb{H}$ (not necessarily with the underlying graph in $\mathcal{F}$) is $k$-universal for $\mathcal{F}$ when any $k$-edge-colored graph with the underlying graph in $\mathcal{F}$ admits a homomorphism to $\mathbb{H}$. We characterize graph classes that admit $k$-universal graphs. For such classes, we establish asymptotically almost tight bounds on the size of the smallest universal graph. For a nonempty graph $G$, the density of $G$ is the maximum ratio of the number of edges to the number of vertices ranging over all nonempty subgraphs of $G$. For a nonempty class $\mathcal{F}$ of graphs, $D(\mathcal{F})$ denotes the density of $\mathcal{F}$, that is the supremum of densities of graphs in $\mathcal{F}$. The main results are the following. The class $\mathcal{F}$ admits $k$-universal graphs for $k\geq2$ if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in $\mathcal{F}$. For any such class, there exists a constant $c$, such that for any $k \geq 2$, the size of the smallest $k$-universal graph is between $k^{D(\mathcal{F})}$ and $ck^{\lceil D(\mathcal{F})\rceil}$. A connection between the acyclic coloring and the existence of universal graphs was first observed by Alon and Marshall (Journal of Algebraic Combinatorics, 8(1):5-13, 1998). One of their results is that for planar graphs, the size of the smallest $k$-universal graph is between $k^3+3$ and $5k^4$. Our results yield that there exists a constant $c$ such that for all $k$, this size is bounded from above by $ck^3$.

math.CO↗

arXiv2016

The Partial Visibility Representation Extension Problem

For a graph $G$, a function $ψ$ is called a \emph{bar visibility representation} of $G$ when for each vertex $v \in V(G)$, $ψ(v)$ is a horizontal line segment (\emph{bar}) and $uv \in E(G)$ iff there is an unobstructed, vertical, $\varepsilon$-wide line of sight between $ψ(u)$ and $ψ(v)$. Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph $G$, a bar visibility representation $ψ$ of $G$, additionally, puts the bar $ψ(u)$ strictly below the bar $ψ(v)$ for each directed edge $(u,v)$ of $G$. We study a generalization of the recognition problem where a function $ψ'$ defined on a subset $V'$ of $V(G)$ is given and the question is whether there is a bar visibility representation $ψ$ of $G$ with $ψ(v) = ψ'(v)$ for every $v \in V'$. We show that for undirected graphs this problem together with closely related problems are \NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.

cs.CG↗

arXiv2016

Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints

We propose two strategies for Presenter in the on-line interval graph coloring games. Specifically, we consider a setting in which each interval is associated with a $d$-dimensional vector of weights and the coloring needs to satisfy the $d$-dimensional bandwidth constraint, and the $k$-cardinality constraint. Such a variant was first introduced by Epstein and Levy and it is a natural model for resource-aware task scheduling with $d$ different shared resources where at most $k$ tasks can be scheduled simultaneously on a single machine. The first strategy forces any on-line interval coloring algorithm to use at least $(5m-3)\frac{d}{\log d + 3}$ different colors on an $m(\frac{d}{k} + \log{d} + 3)$-colorable set of intervals. The second strategy forces any on-line interval coloring algorithm to use at least $\lfloor\frac{5m}{2}\rfloor\frac{d}{\log d + 3}$ different colors on an $m(\frac{d}{k} + \log{d} + 3)$-colorable set of unit intervals.

math.CO↗

arXiv2015

Lower bounds for on-line graph colorings

We propose two strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use $2\log_2 n - 10$ colors, where $n$ is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs graphs that contain neither $C_3$ nor $C_5$ as a subgraph and forces $Ω(\frac{n}{\log n}^\frac{1}{3})$ colors. The best known on-line coloring algorithm for these graphs uses $O(n^{\frac{1}{2}})$ colors.

math.CO↗

arXiv2015

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 $Θ(\log \log N )$. 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 $Θ(2^k )$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.

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