arXiv subjects

Zsolt Tuza

Publications and source records attributed to Zsolt Tuza.

At least 55 records · Page 3Linked to original sources

arXiv2019

Singular Turán numbers and WORM-colorings

A subgraph $H$ of $G$ is \textit{singular} if the vertices of $H$ either have the same degree in $G$ or have pairwise distinct degrees in $G$. The largest number of edges of a graph on $n$ vertices that does not contain a singular copy of $H$ is denoted by $T_S(n,H)$. Caro and Tuza [Theory and Applications of Graphs, 6 (2019), 1--32] obtained the asymptotics of $T_S(n,H)$ for every graph $H$, but determined the exact value of this function only in the case $H=K_3$ and $n\equiv 2$ (mod 4). We determine $T_S(n,K_3)$ for all $n\equiv 0$ (mod 4) and $n\equiv 1$ (mod 4), and also $T_S(n,K_{r+1})$ for large enough $n$ that is divisible by $r$. We also explore the connection to the so-called $H$-WORM colorings (colorings without rainbow or monochromatic copies of $H$) and obtain new results regarding the largest number of edges that a graph with an $H$-WORM coloring can have.

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arXiv2019

Singular Ramsey and Turán numbers

We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $v\in V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $m\geq n$, in every edge 2-coloring of $K_m$, at least one of the color classes contains $F$ as a singular subgraph. In a similar flavor, the singular Turán number Ts$(n,F)$ is defined as the maximum number of edges in a graph of order $n$, which does not contain $F$ as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop methods to estimate Rs$(F)$ and Ts$(n,F)$, present tight asymptotic bounds and exact results.

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arXiv2019

Realization of digraphs in Abelian groups and its consequences

Let $\overrightarrow{G}$ be a directed graph with no component of orderless than~$3$, and let $Γ$ be a finite Abelian group such that $|Γ|\geq 4|V(\overrightarrow{G})|$ or if $|V(\overrightarrow{G})|$ is large enough with respect to an arbitrarily fixed $\varepsilon>0$ then $|Γ|\geq (1+\varepsilon)|V(\overrightarrow{G})|$. We show that there exists an injective mapping $φ$ from $V(\overrightarrow{G})$ to the group $Γ$ such that $\sum_{x\in V(C)}φ(x) = 0$ for every connected component $C$ of $\overrightarrow{G}$, where $0$ is the identity element of $Γ$. Moreover we show some applications of this result to group distance magic labelings.

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arXiv2018

The variety of domination games

Domination game [SIAM J.\ Discrete Math.\ 24 (2010) 979--991] and total domination game [Graphs Combin.\ 31 (2015) 1453--1462] are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than $6/7$ of the order of a graph.

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arXiv2018

Subexponential-time Algorithms for Maximum Independent Set in $P_t$-free and Broom-free Graphs

In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on $P_t$-free graphs, that is, on graphs not containing any induced path on $t$ vertices. So far, polynomial-time algorithms are known only for $t\le 5$ [Lokshtanov et al., SODA 2014, 570--581, 2014], and an algorithm for $t=6$ announced recently [Grzesik et al. Arxiv 1707.05491, 2017]. Here we study the existence of subexponential-time algorithms for the problem: we show that for any $t\ge 1$, there is an algorithm for Maximum Independent Set on $P_t$-free graphs whose running time is subexponential in the number of vertices. Even for the weighted version MWIS, the problem is solvable in $2^{O(\sqrt {tn \log n})}$ time on $P_t$-free graphs. For approximation of MIS in broom-free graphs, a similar time bound is proved. Scattered Set is the generalization of Maximum Independent Set where the vertices of the solution are required to be at distance at least $d$ from each other. We give a complete characterization of those graphs $H$ for which $d$-Scattered Set on $H$-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges): If every component of $H$ is a path, then $d$-Scattered Set on $H$-free graphs with $n$ vertices and $m$ edges can be solved in time $2^{O(|V(H)|\sqrt{n+m}\log (n+m))}$, even if $d$ is part of the input. Otherwise, assuming the Exponential-Time Hypothesis (ETH), there is no $2^{o(n+m)}$-time algorithm for $d$-Scattered Set for any fixed $d\ge 3$ on $H$-free graphs with $n$-vertices and $m$-edges.

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arXiv2018

Partition-crossing hypergraphs

For a finite set $X$, we say that a set $H\subseteq X$ crosses a partition ${\cal P}=(X_1,\dots,X_k)$ of $X$ if $H$ intersects $\min (|H|,k)$ partition classes. If $|H|\geq k$, this means that $H$ meets all classes $X_i$, whilst for $|H|\leq k$ the elements of the crossing set $H$ belong to mutually distinct classes. A set system ${\cal H}$ crosses ${\cal P}$, if so does some $H\in {\cal H}$. The minimum number of $r$-element subsets, such that every $k$-partition of an $n$-element set $X$ is crossed by at least one of them, is denoted by $f(n,k,r)$. The problem of determining these minimum values for $k=r$ was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387--1404]. The present authors determined asymptotically tight estimates on $f(n,k,k)$ for every fixed $k$ as $n\to \infty$ [Graphs Combin., 25 (2009), 807--816]. Here we consider the more general problem for two parameters $k$ and $r$, and establish lower and upper bounds for $f(n,k,r)$. For various combinations of the three values $n,k,r$ we obtain asymptotically tight estimates, and also point out close connections of the function $f(n,k,r)$ to Turán-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs.

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arXiv2017

On Grundy total domination number in product graphs

A longest sequence $(v_1,\ldots,v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, $N(v_i) \setminus \bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $γ_{gr}^{t}(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $γ_{gr}^t(G\times H) \geq γ_{gr}^t(G)γ_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic product we express $γ_{gr}^t(G\circ H)$ in terms of related invariant of the factors and find some explicit formulas for it. For the strong product, lower bounds on $γ_{gr}^t(G \boxtimes H)$ are proved as well as upper bounds for products of paths and cycles. For the Cartesian product we prove lower and upper bounds on the Grundy total domination number when factors are paths or cycles.

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arXiv2017

Domination game on uniform hypergraphs

In this paper we introduce and study the domination game on hypergraphs. This is played on a hypergraph $\mathcal{H}$ by two players, namely Dominator and Staller, who alternately select vertices such that each selected vertex enlarges the set of vertices dominated so far. The game is over if all vertices of $\mathcal{H}$ are dominated. Dominator aims to finish the game as soon as possible, while Staller aims to delay the end of the game. If each player plays optimally and Dominator starts, the length of the game is the invariant `game domination number' denoted by $γ_g(\mathcal{H})$. This definition is the generalization of the domination game played on graphs and it is a special case of the transversal game on hypergraphs. After some basic general results, we establish an asymptotically tight upper bound on the game domination number of $k$-uniform hypergraphs. In the remaining part of the paper we prove that $γ_g(\mathcal{H}) \le 5n/9$ if $\mathcal{H}$ is a 3-uniform hypergraph of order $n$ and does not contain isolated vertices. This also implies the following new result for graphs: If $G$ is an isolate-free graph on $n$ vertices and each of its edges is contained in a triangle, then $γ_g(G) \le 5n/9$.

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arXiv2017

A Note on the Polytope of Bipartite TSP

The main result of this paper is that the polytope of the bipartite TSP is significantly different from that of the general TSP. Comb inequalities are known as facet defining ones in the general case. In the bipartite case, however, many of them are satisfied whenever all degree and subtour elimination constraints are satisfied, {\em i.e.}\ these comb inequalities are not facet defining. The inequalities in question belong to the cases where vertices of one of the two classes occur in less than the half of the intersections of the teeth and the hand. Such side conditions are necessary, as simple example shows that the comb inequality can be violated when each class has vertices in more than the half of the intersections.

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arXiv2017

Grundy dominating sequences and zero forcing sets

In a graph $G$ a sequence $v_1,v_2,\dots,v_m$ of vertices is Grundy dominating if for all $2\le i \le m$ we have $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ and is Grundy total dominating if for all $2\le i \le m$ we have $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. The length of the longest Grundy (total) dominating sequence has been studied by several authors. In this paper we introduce two similar concepts when the requirement on the neighborhoods is changed to $N(v_i)\not\subseteq \cup_{j=1}^{i-1}N[v_j]$ or $N[v_i]\not\subseteq \cup_{j=1}^{i-1}N(v_j)$. In the former case we establish a strong connection to the zero forcing number of a graph, while we determine the complexity of the decision problem in the latter case. We also study the relationships among the four concepts, and discuss their computational complexities.

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arXiv2016

Graph labeling games

We propose the study of many new variants of two-person graph labeling games. Hardly anything has been done in this wide open field so far.

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arXiv2016

Clique Coverings and Claw-free Graphs

Let $\cal C$ be a clique covering for $E(G)$ and let $v$ be a vertex of $G$. The valency of vertex $v$ (with respect to $\cal C$), denoted by $val_{\cal C}(v)$, is the number of cliques in $\cal C$ containing $v$. The local clique cover number of $G$, denoted by $lcc(G)$, is defined as the smallest integer $k$, for which there exists a clique covering for $E(G)$ such that $val_{\cal C}(v)$ is at most $k$, for every vertex $v\in V(G)$. In this paper, among other results, we prove that if $G$ is a claw-free graph, then $lcc(G)+χ(G)\leq n+1$.

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arXiv2016

The minimum number of vertices in uniform hypergraphs with given domination number

The \textit{domination number} $γ(\mathcal{H})$ of a hypergraph $\mathcal{H}=(V(\mathcal{H}),E(\mathcal{H})$ is the minimum size of a subset $D\subset V(\mathcal{H}$ of the vertices such that for every $v\in V(\mathcal{H})\setminus D$ there exist a vertex $d \in D$ and an edge $H\in E(\mathcal{H})$ with $v,d\in H$. We address the problem of finding the minimum number $n(k,γ)$ of vertices that a $k$-uniform hypergraph $\mathcal{H}$ can have if $γ(\mathcal{H})\ge γ$ and $\mathcal{H}$ does not contain isolated vertices. We prove that $$n(k,γ)=k+Θ(k^{1-1/γ})$$ and also consider the $s$-wise dominating and the distance-$l$ dominating version of the problem. In particular, we show that the minimum number $n_{dc}(k,γ, l)$ of vertices that a connected $k$-uniform hypergraph with distance-$l$ domination number $γ$ can have is roughly $\frac{kγl}{2}$

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arXiv2016

Dominating sequences in grid-like and toroidal graphs

A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.

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arXiv2016

Computing all possible graph structures describing linearly conjugate realizations of kinetic systems

In this paper an algorithm is given to determine all possible structurally different linearly conjugate realizations of a given kinetic polynomial system. The solution is based on the iterative search for constrained dense realizations using linear programming. Since there might exist exponentially many different reaction graph structures, we cannot expect to have a polynomial-time algorithm, but we can organize the computation in such a way that polynomial time is elapsed between displaying any two consecutive realizations. The correctness of the algorithm is proved, and possibilities of a parallel implementation are discussed. The operation of the method is shown on two illustrative examples.

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arXiv2016

Bounds on the Game Transversal Number in Hypergraphs

Let $H = (V,E)$ be a hypergraph with vertex set $V$ and edge set $E$ of order $\nH = |V|$ and size $\mH = |E|$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty intersection with every edge of $H$. A vertex hits an edge if it belongs to that edge. The transversal game played on $H$ involves of two players, \emph{Edge-hitter} and \emph{Staller}, who take turns choosing a vertex from $H$. Each vertex chosen must hit at least one edge not hit by the vertices previously chosen. The game ends when the set of vertices chosen becomes a transversal in $H$. Edge-hitter wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The \emph{game transversal number}, $τ_g(H)$, of $H$ is the number of vertices chosen when Edge-hitter starts the game and both players play optimally. We compare the game transversal number of a hypergraph with its transversal number, and also present an important fact concerning the monotonicity of $τ_g$, that we call the Transversal Continuation Principle. It is known that if $H$ is a hypergraph with all edges of size at least~$2$, and $H$ is not a $4$-cycle, then $τ_g(H) \le \frac{4}{11}(\nH+\mH)$; and if $H$ is a (loopless) graph, then $τ_g(H) \le \frac{1}{3}(\nH + \mH + 1)$. We prove that if $H$ is a $3$-uniform hypergraph, then $τ_g(H) \le \frac{5}{16}(\nH + \mH)$, and if $H$ is $4$-uniform, then $τ_g(H) \le \frac{71}{252}(\nH + \mH)$.

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arXiv2015

A Combinatorial Problem Related to Sparse Systems of Equations

Nowadays sparse systems of equations occur frequently in science and engineering. In this contribution we deal with sparse systems common in cryptanalysis. Given a cipher system, one converts it into a system of sparse equations, and then the system is solved to retrieve either a key or a plaintext. Raddum and Semaev proposed new methods for solving such sparse systems. It turns out that a combinatorial MaxMinMax problem provides bounds on the average computational complexity of sparse systems. In this paper we initiate a study of a linear algebra variation of this MaxMinMax problem.

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arXiv2015

$F$-WORM colorings: Results for 2-connected graphs

Given two graphs $F$ and $G$, an $F$-WORM coloring of $G$ is an assignment of colors to its vertices in such a way that no $F$-subgraph of $G$ is monochromatic or rainbow. If $G$ has at least one such coloring, then it is called $F$-WORM colorable and $W^-(G,F)$ denotes the minimum possible number of colors. Here, we consider $F$-WORM colorings with a fixed 2-connected graph $F$ and prove the following three main results: (1) For every natural number $k$, there exists a graph $G$ which is $F$-WORM colorable and $W^-(G,F)=k$; (2) It is NP-complete to decide whether a graph is $F$-WORM colorable; (3) For each $k \ge |V(F)|-1$, it is NP-complete to decide whether a graph $G$ satisfies $W^-(G,F) \le k$. This remains valid on the class of $F$-WORM colorable graphs of bounded maximum degree. For complete graphs $F=K_n$ with $n \ge 3$ we also prove: (4) For each $n \ge 3$ there exists a graph $G$ and integers $r$ and $s$ such that $s \ge r+2$, $G$ has $K_n$-WORM colorings with exactly $r$ and also with $s$ colors, but it admits no $K_n$-WORM colorings with exactly $r+1, \dots, s-1$ colors. Moreover, the difference $s-r$ can be arbitrarily large.

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