arXiv subjects

Grzegorz Gutowski

Publications and source records attributed to Grzegorz Gutowski.

At least 19 recordsLinked to original sources

arXiv2026

How to Construct High Barrycades

Given two positive integers height $h$ and order $n$, the barrycade construction problem asks for a set of $h$ permutations of the integers from $1$ to $n$ such that all the proper partial sums given by these permutations are pairwise distinct. The name barrycade was coined by Richard K. Guy and refers to Barry Cipra, who introduced this kind of arrangement problem. A simple calculation shows that a solution can only exist for $n \ge 2h-2$ and it is conjectured that there always exists a solution for every height $h \ge 2$ and order $n \ge 2h-2$. In this work, for every height $h \ge 1$, we present a construction of a barrycade of height $h$ and order $n = 2h+3$. We also present a randomized heuristic that allows us to find a barrycade of height $h$ and conjectured optimal order $n=2h-2$, for every height $2 \leq h \leq 50$. Thus, we confirm the conjectured optimal order for all heights up to $50$. We also consider a related corral construction problem, where the permutations define a cyclic arrangement. In this setting, for every height $h \ge 1$, we present a construction of a corral of height $h$ and order $n=2h$. A heuristic approach, similar to the one used for barrycades, allows us to find a corral of height $h$ and conjectured optimal order $n=2h-1$, for every height $1 \leq h \leq 50$. We confirm Tomoki Nakamigawa's conjecture on well-dispersed partitions of cyclic groups for the number of parts up to $20$.

math.CO↗

arXiv2026

Online and Incremental Fractional Vertex Cover on Trees

In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.

cs.DS↗

arXiv2026

Alon-Tarsi for hypergraphs

Given a hypergraph $H=(V,E)$, define for every edge $e\in E$ a linear expression with arguments corresponding to the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal is to find a relationship between the Alon-Tarsi number of $p_H$ and the edge density of $H$. We prove that $AT(p_H)=\lceil \mathrm{ed}(H)\rceil+1$ if all the coefficients in $p_H$ are equal to $1$ and the base field has characteristic zero. Our main result is that, over an arbitrary field, if on every edge the coefficients are not all equal, then they can be permuted within the edges so that for the resulting polynomial $p_H^\prime$, $AT(p_H^\prime)\leq 2\lceil \mathrm{ed}(H)\rceil+1$ holds. We conjecture that this bound holds for every hypergraph polynomial without permuting its coefficients. If this were true, then in particular a significant generalization of the famous 1-2-3 Conjecture would follow.

math.CO↗

arXiv2026

Arboricity Nearly Bounds Degeneracy

Arboricity and degeneracy are two fundamental and closely related graph parameters that measure the sparsity of a graph. Every $k$-degenerate graph is $k$-arboric, but some $k$-arboric graphs are only $(2k-1)$-degenerate. However, every maximal $k$-arboric multigraph with $n$ vertices and every maximal $k$-degenerate multigraph with $n$ vertices has exactly $k(n-1)$ edges. These basic observations lead to a natural structural question: How far are $k$-arboric graphs from being $k$-degenerate? We answer this question by showing that: By at most a $(k-1)$-bounded-degree graph apart. More specifically, we prove that a $k$-arboric multigraph admits a $(k,k-1)$-decomposition, that is, its edges can be partitioned into two multisets such that one spans a $k$-degenerate multigraph and the other spans a multigraph with every vertex having degree at most $k-1$. Moreover, we provide a complete characterisation of all possible such decomposition types. Namely, for any integers $k \ge 1$ and $d,h \ge 0$ we show that every $k$-arboric multigraph admits a $(d,h)$-decomposition if and only if $d\geq k$ and $d+h\geq 2k-1$. Our proofs are constructive and we present a polynomial time algorithm that produces such decompositions. By contrast, we show that related decision problems for general graphs (without constraints on the arboricity) are NP-complete.

math.CO↗

arXiv2026

How Close is a Tree to a Euclidean Minimum Spanning Tree?

Let $Γ$ be a straight-line crossing-free drawing of a tree $T$. A \emph{bad pair} in $Γ$ is a pair of non-adjacent vertices of $T$ whose Euclidean distance in $Γ$ is smaller than the length of the longest edge in the path connecting them in~$Γ$. When $Γ$ has no bad pairs, $Γ$ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be \NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For $n$-vertex trees with maximum vertex degree $Δ$, we prove the $Δ^2n\log n$ upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.

cs.CG↗

arXiv2026

Modelling Network Resilience: The Complexity of Some Graph Division Games

Motivated by the controller placement problems in software-defined networks and the fair division principles of classical "cake cutting", we investigate the following two-player zero-sum game. In our model, a defender places a limited number of controllers on graph vertices, while an attacker deletes a limited number of vertices. The defender score is the total number of surviving vertices reachable from any remaining controller. We formalize the computational problems associated with various game dynamics (defender plays first; attacker plays first; players play simultaneously; pure or mixed strategies). We show that these natural problems are $\mathsf{NP}$-complete or $Σ^\mathsf{P}_2$-complete, depending on the specific variant. These hardness results provide limitations for optimal controller placement algorithms under different notions of quality of a solution. Finally, we present structural insights that yield efficient algorithms for restricted graph classes (namely interval graphs and graphs of bounded treewidth).

cs.CC↗

arXiv2026

One-Sided Local Crossing Minimization

Drawing graphs with the minimum number of crossings is a classical problem that has been studied extensively. Many restricted versions of the problem have been considered. For example, bipartite graphs can be drawn such that the two sets in the bipartition of the vertex set are mapped to two parallel lines, and the edges are drawn as straight-line segments. In this setting, the number of crossings depends only on the ordering of the vertices on the two lines. Two natural variants of the problem have been studied. In the one-sided case, the order of the vertices on one of the two lines is given and fixed; in the two-sided case, no order is given. Both cases are important yet NP-hard subproblems in the so-called Sugiyama framework for drawing layered graphs with few crossings. For the one-sided case, Eades and Wormald [Algorithmica 1994] introduced a median heuristic and showed that it has an approximation ratio of 3. In recent years, researchers have focused on a local version of crossing minimization, where the aim is to minimize the maximum number of crossings per edge instead of the total number of crossings. Kobayashi, Okada, and Wolff [SoCG 2025] investigated the complexity of local crossing minimization parameterized by the natural parameter. They conjectured that one-sided local crossing minimization is NP-hard. In this work, we confirm their conjecture by showing that the problem is NP-hard even for forests of high-degree stars. In fact, more strongly, the reduction yields a tight lower bound, which excludes the existence of subexponential-time algorithms assuming the Exponential-Time Hypothesis. In contrast, we present a quadratic-time algorithm for the special case of forests of stars of maximum degree 2. Finally, we provide a median heuristic with a carefully designed tie-breaking scheme and prove that it has an approximation ratio of 3 in the local setting.

cs.DS↗

arXiv2026

Hunting for Directed 2-Spiders

Hons, Klimošová, Kucheriya, Mikšaník, Tkadlec, and Tyomkyn proved that, for every integer $\ell \ge 1$, every directed graph with minimum out-degree at least $3.23 \cdot \ell$ contains a $(2,\ell)$-spider (a $1$-subdivision of the in-star with $\ell$ leaves) as a subgraph. They also conjectured that the bound on the minimum out-degree can be further improved to $2 \ell$. In this note, we confirm their conjecture by showing that every directed graph with minimum out-degree at least $2\ell$ contains a $(2, \ell)$-spider as a subgraph. This result is best possible, as the complete directed graph with $2\ell$ vertices does not contain a $(2,\ell)$-spider.

math.CO↗

arXiv2026

A Note on the Complexity of Directed Clique

For a directed graph $G$, and a linear order $\ll$ on the vertices of $G$, we define backedge graph $G^\ll$ to be the undirected graph on the same vertex set with edge $\{u,w\}$ in $G^\ll$ if and only if $(u,w)$ is an arc in $G$ and $w \ll u$. The directed clique number of a directed graph $G$ is defined as the minimum size of the maximum clique in the backedge graph $G^\ll$ taken over all linear orders $\ll$ on the vertices of $G$. A natural computational problem is to decide for a given directed graph $G$ and a positive integer $t$, if the directed clique number of $G$ is at most $t$. This problem has polynomial algorithm for $t=1$ and is known to be \NP-complete for every fixed $t\ge3$, even for tournaments. In this note we prove that this problem is $Σ^\mathsf{P}_{2}$-complete when $t$ is given on the input.

cs.CC↗

arXiv2025

Mrs. Correct and Majority Colorings

A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for paintability - an on-line analog of the list coloring.

math.CO↗

arXiv2025

A Note on the Complexity of Defensive Domination

In a graph G, a k-attack A is any set of at most k vertices and l-defense D is a set of at most l vertices. We say that defense D counters attack A if each a in A can be matched to a distinct defender d in D with a equal to d or a adjacent to d in G. In the defensive domination problem, we are interested in deciding, for a graph G and positive integers k and l given on input, if there exists an l-defense that counters every possible k-attack on G. Defensive domination is a natural resource allocation problem and can be used to model network robustness and security, disaster response strategies, and redundancy designs. The defensive domination problem is naturally in the complexity class $Σ^P_2$. The problem was known to be NP-hard in general, and polynomial-time algorithms were found for some restricted graph classes. In this note we prove that the defensive domination problem is $Σ^P_2$-complete. We also introduce a natural variant of the defensive domination problem in which the defense is allowed to be a multiset of vertices. This variant is also $Σ^P_2$-complete, but we show that it admits a polynomial-time algorithm in the class of interval graphs. A similar result was known for the original setting in the class of proper interval graphs.

cs.CC↗

arXiv2024

Bounding the Treewidth of Outer $k$-Planar Graphs via Triangulations

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most $k$ other edges. Wood and Telle [New York J. Math., 2007] showed that every outer $k$-planar graph has treewidth at most $3k + 11$ using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer $1$-planar graphs is at most $3$, which is tight. In this paper, we improve the general upper bound to $1.5k + 2$ and give a tight bound of $4$ for $k = 2$. We also establish a lower bound: we show that, for every even $k$, there is an outer $k$-planar graph with treewidth $k+2$. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer $k$-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of $k + 2$ for the separation number of outer $k$-planar graphs, improving an upper bound of $2k + 3$ by Chaplick et al. [GD 2017]. We also consider outer min-$k$-planar graphs, a generalization of outer $k$-planar graphs, where we achieve smaller improvements.

cs.DM↗

arXiv2024

First-Fit Coloring of Forests in Random Arrival Model

We consider a graph coloring algorithm that processes vertices in order taken uniformly at random and assigns colors to them using First-Fit strategy. We show that this algorithm uses, in expectation, at most $(1 + o(1))\cdot \ln n \,/\, \ln\ln n$ different colors to color any forest with $n$ vertices. We also construct a family of forests that shows that this bound is best possible.

cs.DM↗

arXiv2023

Coloring and Recognizing Directed Interval Graphs

A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph $G$, an interval $u$ receives a lower (different) color than an interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a $\min \{ω(G), λ(G)+1 \}$-approximation algorithm, where $ω(G)$ is the size of a largest clique and $λ(G)$ is the length of a longest directed path in $G$. For the subclass of \emph{bidirectional interval graphs} (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call \emph{containment interval graphs}. In such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$, and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.

cs.DM↗

arXiv2022

Coloring Mixed and Directional Interval Graphs

A mixed graph has a set of vertices, a set of undirected egdes, and a set of directed arcs. A proper coloring of a mixed graph $G$ is a function $c$ that assigns to each vertex in $G$ a positive integer such that, for each edge $uv$ in $G$, $c(u) \ne c(v)$ and, for each arc $uv$ in $G$, $c(u) < c(v)$. For a mixed graph $G$, the chromatic number $χ(G)$ is the smallest number of colors in any proper coloring of $G$. A directional interval graph is a mixed graph whose vertices correspond to intervals on the real line. Such a graph has an edge between every two intervals where one is contained in the other and an arc between every two overlapping intervals, directed towards the interval that starts and ends to the right. Coloring such graphs has applications in routing edges in layered orthogonal graph drawing according to the Sugiyama framework; the colors correspond to the tracks for routing the edges. We show how to recognize directional interval graphs, and how to compute their chromatic number efficiently. On the other hand, for mixed interval graphs, i.e., graphs where two intersecting intervals can be connected by an edge or by an arc in either direction arbitrarily, we prove that computing the chromatic number is NP-hard.

cs.DM↗

arXiv2021

Graph Polynomials and Group Coloring of Graphs

Let $Γ$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $Γ$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow Γ$, there exists a vertex coloring $c$ by the elements of $Γ$ such that $c(y)-c(x)\neq \ell(e)$, for every edge $e=xy$ (oriented from $x$ to $y$). Langhede and Thomassen proved recently that every planar graph on $n$ vertices has at least $2^{n/9}$ different $\mathbb{Z}_5$-colorings. By using a different approach based on graph polynomials, we extend this result to $K_5$-minor-free graphs in the more general setting of field coloring. More specifically, we prove that every such graph on $n$ vertices is $\mathbb{F}$-$5$-choosable, whenever $\mathbb{F}$ is an arbitrary field with at least $5$ elements. Moreover, the number of colorings (for every list assignment) is at least $5^{n/4}$.

math.CO↗

arXiv2021

On a Problem of Steinhaus

Let $N$ be a positive integer. A sequence $X=(x_1,x_2,\ldots,x_N)$ of points in the unit interval $[0,1)$ is piercing if $\{x_1,x_2,\ldots,x_n\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. In 1958 Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most $74$ elements. This was later improved to the best possible value of $17$ by Warmus, and independently by Berlekamp and Graham. In this paper we study a more general variant of piercing sequences. Let $f(n)\geq n$ be an infinite nondecreasing sequence of positive integers. A sequence $X=(x_1,x_2,\ldots,x_{f(N)})$ is $f$-piercing if $\{x_1,x_2,\ldots,x_{f(n)}\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. A special case of $f(n)=n+d$, with $d$ a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each $d\geq 0$, the maximum length of any $(n+d)$-piercing sequence is finite. Expressing this maximum length as $s(d)+d$, they obtained an exponential upper bound on the function $s(d)$, which was later improved to $s(d)=O(d^3)$ by Graham and Levy. Recently, Konyagin proved that $2d\leqslant s(d)< 200d$ holds for all sufficiently big $d$. Using a different technique based on the Farey fractions and stick-breaking games, we prove here that the function $s(d)$ satisfies $\left\lfloor{}c_1d\right\rfloor{}\leqslant s(d)\leqslant c_2d+o(d)$, where $c_1=\frac{\ln 2}{1-\ln 2}\approx2.25$ and $c_2=\frac{1+\ln2}{1-\ln2}\approx5.52$. We also prove that there exists an infinite $f$-piercing sequence with $f(n)= γn+o(n)$ if and only if $γ\geq\frac{1}{\ln 2}\approx 1.44$.

math.NT↗

arXiv2020

Online Coloring of Short Intervals

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in $[1,σ]$. For $σ=1$ it is the class of unit interval graphs, and for $σ=\infty$ the class of all interval graphs. Our focus is on intermediary classes. We present a $(1+σ)$-competitive algorithm, which beats the state of the art for $1 < σ< 2$, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than $5/3$-competitive for any $σ>1$, nor better than $7/4$-competitive for any $σ>2$, and that no algorithm beats the $5/2$ asymptotic competitive ratio for all, arbitrarily large, values of $σ$. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than $2$.

math.CO↗