arXiv subjects

Ladislav Stacho

Publications and source records attributed to Ladislav Stacho.

At least 19 recordsLinked to original sources

arXiv2025

A Note on Large Degenerate Induced Subgraphs in Sparse Graphs

Given a graph $G$ and a non-negative integer $d$ let $\alpha_d(G)$ be the order of a largest induced $d$-degenerate subgraph of $G$. We prove that for any pair of non-negative integers $k>d$, if $G$ is a $k$-degenerate graph, then $\alpha_d(G) \geq \max\{ \frac{(d+1)n}{k+d+1}, n - \alpha_{k-d-1}(G)\}$. For $k$-degenerate graphs this improves a more general lower bound of Alon, Kahn, and Seymour. By modifying our argument we obtain improved lower bound on $\alpha_d(G)$ for graphs of bounded genus. This extends earlier work on degenerate subgraphs of planar graphs.

math.CO↗

arXiv2024

Bipartite graphs are $(\frac{4}{5}-\varepsilon) \frac{\Delta}{\log \Delta}$-choosable

Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $\Delta$, then the choosability (or list chromatic number) of $G$ satisfies $\chi_{\ell}(G) = O \left ( \log \Delta \right )$. Currently, the best known upper bound for $\chi_{\ell}(G)$ is $(1 + o(1)) \frac{\Delta}{\log \Delta}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $\Delta$ satisfies $\chi_{\ell}(G) < (\frac{4}{5} -\varepsilon) \frac{\Delta}{\log \Delta}$. This improved upper bound suggests that list coloring is fundamentally different for bipartite graphs than for triangle-free graphs and hence gives a step toward solving the conjecture of Alon and Krivelevich.

math.CO↗

arXiv2024

A cornering strategy for synchronizing a DFA

This paper considers the existence of short synchronizing words in deterministic finite automata (DFAs). We define two general strategies for generating synchronizing words, and we show that each of these strategies can be applied if and only if a DFA is synchronizable. Furthermore, we show that if a synchronizable DFA is well-structured, then our strategies generate short synchronizing words. The first of our strategies, called the cornering strategy, takes advantage of states in a DFA with properties similar to those of a polytope vertex. The second of our strategies, similar to the cornering strategy and called the $f$-ordered strategy, takes advantage of a partial order defined on the states of a DFA. We apply our cornering strategy to the class of difference DFAs, whose states form subsets of $\mathbb R^d$ and whose input symbols correspond to translation vectors between states. We show that difference DFAs share many similarities with aperiodic DFAs, and in particular, a difference DFA $M$ has a synchronizing word if and only if it has a universally reachable state. Using the cornering strategy, we also show that under certain conditions, such an $n$-state DFA $M$ has a synchronizing word of length at most $(n-1)^2$ and thereby satisfies \v{C}ern\'y's conjecture. Using the $f$-ordered strategy, we also show that a synchronizable DFA whose states have a certain partial order that is preserved by a set of short words also has a short synchronizing word, and we consider several consequences of this result. Finally, we consider how the cornering strategy can be applied to the problem of synchronizing the product of two DFAs $M_1, M_2$ that share a common alphabet, and we show that the product $M_1 \times M_2$ often has a synchronizing word that is subquadratic in the number of states of $M_1 \times M_2$.

cs.FL↗

arXiv2023

Oriented Colouring Graphs of Bounded Degree and Degeneracy

This paper considers upper bounds on the oriented chromatic number $\chi_o(G)$, of an oriented graph $G$ in terms of its $2$-dipath chromatic number $\chi_2(G)$, degeneracy $d(G)$, and maximum degree $\Delta(G)$. In particular, we show that for all graphs $G$ with $\chi_2(G) \leq k$ where $k \geq 2$ and $d(G) \leq t$ where $t \geq \log_2(k)$, $\chi_o(G) = 33/10(k t^2 2^t)$. This improves an upper bound of MacGillivray, Raspaud, and Swartz of the form $\chi_o(G) \leq 2^{\chi_2(G)} -1$ to a polynomial upper bound for many classes of graphs, in particular, those with bounded degeneracy. Additionally, we asymptotically improve bounds for the oriented chromatic number in terms of maximum degree and degeneracy. For instance, we show that $\chi_o(G) \leq (2\ln2 +o(1))\Delta^2 2^\Delta$ for all graphs, and $\chi_o(G) \leq (2+o(1))\Delta d 2^d$ for graphs where degeneracy grows sublinearly in maximum degree. Here the asypmtotics are in $\Delta$. The former improves the asymptotics of a results by Kostochka, Sopena, and Zhu \cite{kostochka1997acyclic}, while the latter improves the asymptotics of a result by Aravind and Subramanian \cite{aravind2009forbidden}. Both improvements are by a constant factor.

math.CO↗

arXiv2022

Hamiltonicity of covering graphs of trees

In this paper, we consider covering graphs obtained by lifting a tree with a loop at each vertex as a voltage graph over a cyclic group. We generalize a tool of Hell, Nishiyama, and Stacho, known as the billiard strategy, for constructing Hamiltonian cycles in the covering graphs of paths. We show that our extended tool can be used to provide new sufficient conditions for the Hamiltonicity of covering graphs of trees that are similar to those of Batagelj and Pisanski and of Hell, Nishiyama, and Stacho. Next, we focus specifically on covering graphs obtained from trees lifted as voltage graphs over cyclic groups $\mathbb Z_p$ of large prime order $p$. We prove that for a given reflexive tree $T$ whose edge labels are assigned uniformly at random from a finite set, the corresponding lift is almost surely Hamiltonian for a large enough prime-ordered cyclic group $\mathbb Z_p$. Finally, we show that if a reflexive tree $T$ is lifted over a group $\mathbb Z_p$ of a large prime order, then for any assignment of nonzero elements of $\mathbb Z_p$ to the edges of $T$, the corresponding cover of $T$ has a large circumference.

math.CO↗

arXiv2021

Robust Connectivity of Graphs on Surfaces

Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Throughout recent decades, this problem has received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks. Recently, a similar but stronger notion was defined by Bradshaw, Masa\v{r}\'ik, and Stacho [Flexible List Colorings in Graphs with Special Degeneracy Conditions, ISAAC 2020]. They introduced a new invariant for a graph $G$, called the robust connectivity and written $\kappa_\rho(G)$, defined as the minimum value $\frac{|R \cap \Lambda (T)|}{|R|}$ taken over all nonempty subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to maximize $|R \cap \Lambda(T)|$. Large robust connectivity was originally used to show flexible choosability in non-regular graphs. In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of $\Omega(\gamma^{-\frac{1}{r}})$ for the robust connectivity of $r$-connected graphs of Euler genus $\gamma$. Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman [A conjecture on planar graphs, 1979], which states that every planar graph on $n$ vertices contains an induced forest of size at least $n/2$.

math.CO↗

arXiv2021

From one to many rainbow Hamiltonian cycles

Given a graph $G$ and a family $\mathcal{G} = \{G_1,\ldots,G_n\}$ of subgraphs of $G$, a transversal of $\mathcal{G}$ is a pair $(T,\phi)$ such that $T \subseteq E(G)$ and $\phi: T \rightarrow [n]$ is a bijection satisfying $e \in G_{\phi(e)}$ for each $e \in T$. We call a transversal Hamiltonian if $T$ corresponds to the edge set of a Hamiltonian cycle in $G$. We show that, under certain conditions on the maximum degree of $G$ and the minimum degrees of the $G_i \in \mathcal{G}$, for every $\mathcal{G}$ which contains a Hamiltonian transversal, the number of Hamiltonian transversals contained in $\mathcal{G}$ is bounded below by a function of $G$'s maximum degree. This generalizes a theorem of Thomassen stating that, for $m \geq 300$, no $m$-regular graph is uniquely Hamiltonian. We also extend Joos and Kim's recent result that, if $G = K_n$ and each $G_i \in \mathcal{G}$ has minimum degree at least $\frac{n}{2}$, then $\mathcal{G}$ has a Hamiltonian transversal: we show that, in this setting, $\mathcal{G}$ has exponentially many Hamiltonian transversals. Finally, we prove analogues of both of these theorems for transversals which form perfect matchings of $G$.

math.CO↗

arXiv2020

Flexible List Colorings in Graphs with Special Degeneracy Conditions

For a given $\varepsilon > 0$, we say that a graph $G$ is $\varepsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of color lists of size $k$ on $V(G)$, if a preferred color from a list is requested at any set $R$ of vertices, then at least $\varepsilon |R|$ of these requests are satisfied by some $L$-coloring. We consider the question of flexible choosability in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree $\Delta$ that are $\varepsilon$-flexibly $\Delta$-choosable for some $\varepsilon = \varepsilon(\Delta) > 0$, which answers a question of Dvo\v{r}\'ak, Norin, and Postle [List coloring with requests, JGT 2019]. In particular, we show that for any $\Delta\geq 3$, any graph of maximum degree $\Delta$ that is not isomorphic to $K_{\Delta+1}$ is $\frac{1}{6\Delta}$-flexibly $\Delta$-choosable. Our fraction of $\frac{1}{6 \Delta}$ is within a constant factor of being the best possible. We also show that graphs of treewidth $2$ are $\frac{1}{3}$-flexibly $3$-choosable, answering a question of Choi et al.~[arXiv 2020], and we give conditions for list assignments by which graphs of treewidth $k$ are $\frac{1}{k+1}$-flexibly $(k+1)$-choosable. We show furthermore that graphs of treedepth $k$ are $\frac{1}{k}$-flexibly $k$-choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well-understood class of exceptions, 3-connected non-regular graphs of maximum degree $\Delta$ are flexibly $(\Delta - 1)$-degenerate.

math.CO↗

arXiv2020

On the cop number of graphs of high girth

We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth $g$ and minimum degree $δ$ is at least $\tfrac{1}{g}(δ- 1)^{\lfloor \frac{g-1}{4}\rfloor}$. We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent $\tfrac{1}{4}g$ in this lower bound cannot be improved to $(\tfrac{1}{4}+\varepsilon)g$, we are also able to prove that it cannot be increased beyond $\frac{3}{8}g$. This is established by considering a certain family of Ramanujan graphs. In our proof of this bound, we also show that the "weak" Meyniel's conjecture holds for expander graph families of bounded degree.

math.CO↗

arXiv2019

Constructing Geometric Graphs of Cop Number Three

The game of cops and robbers is a pursuit game on graphs where a set of agents, called the cops try to get to the same position of another agent, called the robber. Cops and robbers has been studies on several classes of graphs including geometrically represented graphs. For example, it has been shown that string graphs, including geometric graphs, have cop number at most 15. On the other hand, little is known about geometric graphs of any cop number less than 15 and there is only one example of a geometric graph of cop number three that has as many as 1440 vertices. In this paper we present a construction for subdividing planar graphs of maximum degree $\le 5$ into geometric planar graphs of at least the same cop number. Indeed, our construction shows that there are infinitely many planar geometric graphs of cop number three. We also present another construction that consists in clique substitutions alongside subdividing the edges in a planar graph of maximum degree $\le 9$, resulting in geometric, but not necessarily planar, graphs of at least the same cop number as the starting graphs.

math.CO↗

arXiv2018

Cops and Robbers on Graphs with a Set of Forbidden Induced Subgraphs

It is known that the class of all graphs not containing a graph $H$ as an induced subgraph is cop-bounded if and only if $H$ is a forest whose every component is a path. In this study, we characterize all sets $\mathscr{H}$ of graphs with some $k\in \mathbb{N}$ bounding the diameter of members of $\mathscr{H}$ from above, such that $\mathscr{H}$-free graphs, i.e. graphs with no member of $\mathscr{H}$ as an induced subgraph, are cop-bounded. This, in particular, gives a characterization of cop-bounded classes of graphs defined by a finite set of connected graphs as forbidden induced subgraphs. Furthermore, we extend our characterization to the case of cop-bounded classes of graphs defined by a set $\mathscr{H}$ of forbidden graphs such that there is $k\in\mathbb{N}$ bounding the diameter of components of members of $\mathscr{H}$ from above.

math.CO↗

arXiv2018

Traversal with Enumeration of Geometric Graphs in Bounded Space

In this paper, we provide an algorithm for traversing geometric graphs which visits all vertices, and reports every vertex and edge exactly once. To achieve this, we combine a given geometric graph $G$ with the integer lattice, seen as a graph, in such a way that the resulting hypothetical graph can be traversed using the algorithm in \cite{Chavez}. To overcome the problem with hypothetical vertices and edges, we develop an algorithm for visiting any $k$th neighborhood of a vertex in a graph straight-line drawn in the plane using $O(\log k)$ memory. The memory needed to complete the traversal of a geometric graph then turns out to depend on the maximum ratio of the graph distance and Euclidean distance for pairs of distinct vertices of $G$ at Euclidean distance greater than one and less than $2\sqrt{2}$.

math.CO↗

arXiv2017

Weak Coverage of a Rectangular Barrier

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak barrier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (MinMax). We give an O(n^{3/2}) time algorithm for the MinNum problem for sensors of diameter 1 that are initially placed at integer positions; in contrast we show that the problem is NP-hard even for sensors of diameter 2 that are initially placed at integer positions. We show that the MinSum problem is solvable in O(n log n) time for homogeneous range sensors in arbitrary initial positions, while it is NP-hard for heterogeneous sensor ranges. Finally, we prove that even very restricted homogeneous versions of the MinMax problem are NP-hard.

cs.DC↗

arXiv2016

Combinatorial RNA Design Designability and Structure-Approximating Algorithm in Watson-Crick and Nussinov-Jacobson Energy Models

We consider the Combinatorial RNA Design problem, a minimal instance of RNA design where one must produce an RNA sequence that adopts a given secondary structure as its minimal free-energy structure. We consider two free-energy models where the contributions of base pairs are additive and independent: the purely combinatorial Watson-Crick model, which only allows equally-contributing A -- U and C -- G base pairs, and the real-valued Nussinov-Jacobson model, which associates arbitrary energies to A -- U, C -- G and G -- U base pairs. We first provide a complete characterization of designable structures using restricted alphabets and, in the four-letter alphabet, provide a complete characterization for designable structures without unpaired bases. When unpaired bases are allowed, we characterize extensive classes of (non-)designable structures, and prove the closure of the set of designable structures under the stutter operation. Membership of a given structure to any of the classes can be tested in $Θ$(n) time, including the generation of a solution sequence for positive instances. Finally, we consider a structure-approximating relaxation of the design, and provide a $Θ$(n) algorithm which, given a structure S that avoids two trivially non-designable motifs, transforms S into a designable structure constructively by adding at most one base-pair to each of its stems.

q-bio.QM↗

arXiv2015

Combinatorial RNA Design: Designability and Structure-Approximating Algorithm

In this work, we consider the Combinatorial RNA Design problem, a minimal instance of the RNA design problem which aims at finding a sequence that admits a given target as its unique base pair maximizing structure. We provide complete characterizations for the structures that can be designed using restricted alphabets. Under a classic four-letter alphabet, we provide a complete characterization of designable structures without unpaired bases. When unpaired bases are allowed, we provide partial characterizations for classes of designable/undesignable structures, and show that the class of designable structures is closed under the stutter operation. Membership of a given structure to any of the classes can be tested in linear time and, for positive instances, a solution can be found in linear time. Finally, we consider a structure-approximating version of the problem that allows to extend bands (helices) and, assuming that the input structure avoids two motifs, we provide a linear-time algorithm that produces a designable structure with at most twice more base pairs than the input structure.

q-bio.QM↗

arXiv2014

Pattern overlap implies runaway growth in hierarchical tile systems

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap. This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly *and" avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.

cs.CG↗

arXiv2013

The robber strikes back

We consider the new game of Cops and Attacking Robbers, which is identical to the usual Cops and Robbers game except that if the robber moves to a vertex containing a single cop, then that cop is removed from the game. We study the minimum number of cops needed to capture a robber on a graph $G$, written $cc(G)$. We give bounds on $cc(G)$ in terms of the cop number of $G$ in the classes of bipartite graphs and diameter two, $K_{1,m}$-free graphs.

math.CO↗

arXiv2011

Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

Given a set $P$ of $n$ points in the plane, we solve the problems of constructing a geometric planar graph spanning $P$ 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set $P$, respectively. First, we construct in $O(n\log{n})$ time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in $O(n\log n)$ time a 2-edge connected geometric planar graph spanning $P$ with max edge length bounded by $\sqrt{5}$ times the optimal, assuming that the set $P$ forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in $O(n^2)$ time. We also show that for $k \in O(\sqrt{n})$, there exists a set $P$ of $n$ points in the plane such that even though the Unit Disk Graph spanning $P$ is $k$-vertex connected, there is no 2-edge connected geometric planar graph spanning $P$ even if the length of its edges is allowed to be up to 17/16.

cs.DM↗