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Zsolt Tuza

Publications and source records attributed to Zsolt Tuza.

71 records · Page 4Linked to original sources
arXiv2015

Induced cycles in triangle graphs

The triangle graph of a graph $G$, denoted by ${\cal T}(G)$, is the graph whose vertices represent the triangles ($K_3$ subgraphs) of $G$, and two vertices of ${\cal T}(G)$ are adjacent if and only if the corresponding triangles share an edge. In this paper, we characterize graphs whose triangle graph is a cycle and then extend the result to obtain a characterization of $C_n$-free triangle graphs. As a consequence, we give a forbidden subgraph characterization of graphs $G$ for which ${\cal T}(G)$ is a tree, a chordal graph, or a perfect graph. For the class of graphs whose triangle graph is perfect, we verify a conjecture of the third author concerning packing and covering of triangles.

math.CO↗
arXiv2015

$K_3$-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum

A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get precisely two colors. We study graphs $G$ which admit at least one such coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014) 161--173] who asked whether every such graph has a $K_3$-WORM coloring with two colors. In fact for every integer $k\ge 3$ there exists a $K_3$-WORM colorable graph in which the minimum number of colors is exactly $k$. There also exist $K_3$-WORM colorable graphs which have a $K_3$-WORM coloring with two colors and also with $k$ colors but no coloring with any of $3,\dots,k-1$ colors. We also prove that it is NP-hard to determine the minimum number of colors and NP-complete to decide $k$-colorability for every $k \ge 2$ (and remains intractable even for graphs of maximum degree 9 if $k=3$). On the other hand, we prove positive results for $d$-degenerate graphs with small $d$, also including planar graphs. Moreover we point out a fundamental connection with the theory of the colorings of mixed hypergraphs. We list many open problems at the end.

math.CO↗
arXiv2015

Transversal designs and induced decompositions of graphs

We prove that for every complete multipartite graph $F$ there exist very dense graphs $G_n$ on $n$ vertices, namely with as many as ${n\choose 2}-cn$ edges for all $n$, for some constant $c=c(F)$, such that $G_n$ can be decomposed into edge-disjoint induced subgraphs isomorphic to~$F$. This result identifies and structurally explains a gap between the growth rates $O(n)$ and $Ω(n^{3/2})$ on the minimum number of non-edges in graphs admitting an induced $F$-decomposition.

math.CO↗
arXiv2014

The Disjoint Domination Game

We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the $(2:1)$ biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted variant of the non-biased game we prove that the maker can win on every graph without isolated vertices.

math.CO↗
arXiv2013

Minimum order of graphs with given coloring parameters

A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $φ:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number $χ(G)$ and achromatic number $ψ(G)$, respectively), and the Grundy number $Γ(G)$ defined as the largest $k$ admitting a complete coloring $φ$ with exactly $k$ colors such that every vertex $v\in V$ of color $φ(v)$ has a neighbor of color $i$ for all $1\le i<φ(v)$. The inequality chain $χ(G)\le Γ(G)\le ψ(G)$ obviously holds for all graphs $G$. A triple $(f,g,h)$ of positive integers at least 2 is called realizable if there exists a connected graph $G$ with $χ(G)=f$, $Γ(G)=g$, and $ψ(G)=h$. Chartrand et al. (A note on graphs with prescribed complete coloring numbers, J. Combin. Math. Combin. Comput. LXXIII (2010) 77-84) found the list of realizable triples. In this paper we determine the minimum number of vertices in a connected graph with chromatic number $f$, Grundy number $g$, and achromatic number $h$, for all realizable triples $(f,g,h)$ of integers. Furthermore, for $f=g=3$ we describe the (two) extremal graphs for each $h \geq 6$. For $h=4$ and $5$, there are more extremal graphs, their description is contained as well.

cs.DM↗
arXiv2013

Uniformly resolvable decompositions of $K_v$ into $P_3$ and $K_3$ graphs

In this paper we consider the uniformly resolvable decompositions of the complete graph $K_v$, or the complete graph minus a 1-factor as appropriate, into subgraphs such that each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes are either $P_3$ or $K_3$.

math.CO↗
arXiv2013

Approximability of the upper chromatic number of hypergraphs

A C-coloring of a hypergraph ${\cal H}=(X,{\cal E})$ is a vertex coloring $φ: X\to {\mathbb{N}}$ such that each edge $E\in{\cal E}$ has at least two vertices with a common color. The related parameter $\overlineχ({\cal H})$, called the upper chromatic number of ${\cal H}$, is the maximum number of colors can be used in a C-coloring of ${\cal H}$. A hypertree is a hypergraph which has a host tree $T$ such that each edge $E\in {\cal E}$ induces a connected subgraph in $T$. Notations $n$ and $m$ stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish guaranteed polynomial-time approximation ratios for the difference $n-\overlineχ({\cal H})$, which is $2+2 \ln (2m)$ on hypergraphs in general, and $1+ \ln m$ on hypertrees. The latter ratio is essentially tight as we show that $n-\overlineχ({\cal H})$ cannot be approximated within $(1-ε) \ln m$ on hypertrees (unless ${\sf NP} \subseteq {\sf DTIME} (n^{{\cal O}(log\;log\; n)})$). Furthermore, $\overlineχ({\cal H})$ does not have ${\cal O}(n^{1-ε})$-approximation and cannot be approximated within additive error $o(n)$ on the class of hypertrees (unless ${\sf P}={\sf NP}$).

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arXiv2013

Total Transversals and Total Domination in Uniform Hypergraphs

The first three authors [European J. Combin. 33 (2012), 62--71] established a relationship between the transversal number and the domination number of uniform hypergraphs. In this paper, we establish a relationship between the total transversal number and the total domination number of uniform hypergraphs. We prove tight asymptotic upper bounds on the total transversal number in terms of the number of vertices, the number of edges, and the edge size.

math.CO↗
arXiv2013

Maximum uniformly resolvable decompositions of $K_v$ and $K_v - I$ into 3-stars and 3-cycles

Let $K_v$ denote the complete graph of order $v$ and $K_v - I$ denote $K_v$ minus a 1-factor. In this article we investigate uniformly resolvable decompositions of $K_v$ and $K_v-I$ into $r$ classes containing only copies of $3$-stars and $s$ classes containing only copies of $3$-cycles. We completely determine the spectrum in the case where the number of resolution classes of 3-stars is maximum.

math.CO↗
arXiv2013

Minimum Number of Affine Simplexes of Given Dimension

In this paper we formulate and solve extremal problems in the d-dimensional Euclidean space and further in hypergraphs, originating from problems in stoichiometry and elementary linear algebra. The notion of affine simplex is the bridge between the original problems and the presented extremal theorem on set systems. A function related to Sperners theorem and the YBLM inequality is also considered and its relation to hypergraph Turan problems is discussed.

math.CO↗
arXiv2013

Turán numbers and batch codes

Combinatorial batch codes provide a tool for distributed data storage, with the feature of keeping privacy during information retrieval. Recently, Balachandran and Bhattacharya observed that the problem of constructing such uniform codes in an economic way can be formulated as a Turán-type question on hypergraphs. Here we establish general lower and upper bounds for this extremal problem, and also for its generalization where the forbidden family consists of those $r$-uniform hypergraphs $H$ which satisfy the condition $k\ge |E(H)|> |V(H)|+q$ (for $k>q+r$ and $q> -r$ fixed). We also prove that, in the given range of parameters, the considered Turán function is asymptotically equal to the one restricted to $|E(H)|=k$, studied by Brown, Erdős and T. Sós. Both families contain some $r$-partite members --- often called the `degenerate case', characterized by the equality $\lim_{n\to\infty} \ex(n,\cF)/n^r=0$ --- and therefore their exact order of growth is not known.

math.CO↗
arXiv2013

Speeding up Deciphering by Hypergraph Ordering

The "Gluing Algorithm" of Semaev [Des.\ Codes Cryptogr.\ 49 (2008), 47--60] --- that finds all solutions of a sparse system of linear equations over the Galois field $GF(q)$ --- has average running time $O(mq^{\max \left\vert \cup_{1}^{k}X_{j}\right\vert -k}), $ where $m$ is the total number of equations, and $\cup_{1}^{k}X_{j}$ is the set of all unknowns actively occurring in the first $k$ equations. Our goal here is to minimize the exponent of $q$ in the case where every equation contains at most three unknowns. %Applying hypergraph-theoretic methods we prove The main result states that if the total number $\left\vert \cup_{1}^{m}X_{j}\right\vert$ of unknowns is equal to $m$, then the best achievable exponent is between $c_1m$ and $c_2m$ for some positive constants $c_1$ and $c_2.$

cs.CR↗
arXiv2012

Bin Packing/Covering with Delivery: Some variations, theoretical results and efficient offline algorithms

In the recent paper \cite{BDT10} we introduced a new problem that we call Bin Packing/Covering with Delivery, or BP/CD for short. Mainly we mean under this expression that we look for not only a good, but a "good and fast" packing or covering. In that paper we mainly dealt with only one possible online BP/CD model, and proposed a new method that we call the Evolution of Algorithms. In case of such methods a neighborhood structure is defined among algorithms, and using a metaheuristic (for example simulated annealing) in some sense the best algorithm is chosen to solve the problem. Now we turn to investigate the offline case. We define several ways to treat such a BP/CD problem, although we investigate only one of them here. For the analysis, a novel view on "offline optimum" is introduced, which appears to be relevant concerning all problems where a final solution is ordering-dependent. We prove that if the item sizes are not allowed to be arbitrarily close to zero, then an optimal offline solution can be found in polynomial time. On the other hand, for unrestricted problem instances, no polynomial-time algorithm can achieve an approximation ratio better than 6/7 if $P\ne NP$.

cs.DS↗
arXiv2011

List colorings of $K_5$-minor-free graphs with special list assignments

A {\it list assignment} $L$ of a graph $G$ is a function that assigns a set (list) $L(v)$ of colors to every vertex $v$ of $G$. Graph $G$ is called {\it $L$-list colorable} if it admits a vertex coloring $ϕ$ such that $ϕ(v)\in L(v)$ for all $v\in V(G)$ and $ϕ(v)\not=ϕ(w)$ for all $vw\in E(G)$. The following question was raised by Bruce Richter. Let $G$ be a planar, 3-connected graph that is not a complete graph. Denoting by $d(v)$ the degree of vertex $v$, is $G$ $L$-list colorable for every list assignment $L$ with $|L(v)|=\min \{d(v), 6\}$ for all $v\in V(G)$? More generally, we ask for which pairs $(r,k)$ the following question has an affirmative answer. Let $r$ and $k$ be integers and let $G$ be a $K_5$-minor-free $r$-connected graph that is not a Gallai tree (i.e., at least one block of $G$ is neither a complete graph nor an odd cycle). Is $G$ $L$-list colorable for every list assignment $L$ with $|L(v)|=\min\{d(v),k\}$ for all $v\in V(G)$? We investigate this question by considering the components of $G[S_k]$, where $S_k:=\{v\in V(G) | d(v)<k\}$ is the set of vertices with small degree in $G$. We are especially interested in the minimum distance $d(S_k)$ in $G$ between the components of $G[S_k]$.

math.CO↗
arXiv2011

Finding weakly reversible realizations of chemical reaction networks using optimization

An algorithm is given in this paper for the computation of dynamically equivalent weakly reversible realizations with the maximal number of reactions, for chemical reaction networks (CRNs) with mass action kinetics. The original problem statement can be traced back at least 30 years ago. The algorithm uses standard linear and mixed integer linear programming, and it is based on elementary graph theory and important former results on the dense realizations of CRNs. The proposed method is also capable of determining if no dynamically equivalent weakly reversible structure exists for a given reaction network with a previously fixed complex set.

math.DS↗
arXiv2004

The Cover Pebbling Number of Graphs

A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of the graph we can eventually put a pebble on every vertex simultaneously. We find the cover pebbling numbers of trees and some other graphs. We also consider the more general problem where (possibly different) given numbers of pebbles are required for the vertices.

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