arXiv subjects

Zoltán Király

Publications and source records attributed to Zoltán Király.

14 recordsLinked to original sources

arXiv2026

Light-induced nonadiabatic photodissociation of the NaH molecule including electron-rotation coupling

It is well established that electronic conical intersections (CIs) in molecular systems can be induced by laser light, even in diatomic molecules. The emergence of these light-induced degeneracies leads to strong coupling among electronic, vibrational, and photonic modes, which significantly influences ultrafast nuclear dynamics. In this work, we perform pump-probe numerical simulations on the NaH molecule, considering the first three singlet electronic states- (X1Σ+(X), A1Σ+(A) and B1Π(B)) -and including several light- induced degeneracies in the theoretical model. To elucidate the ultrafast molecular dynamics, the combined effects of multiple light-induced nonadiabatic couplings and rotational motion of the nuclei, together with the situation when the electronic angular momentum projected onto the diatomic axis couples with the angular momentum of the nuclei has been studied. We then calculate key dynamical observables such as dissociation probabilities, kinetic energy release spectra, and angular distributions of the photofragments within and above the linear regime.

physics.atm-clus↗

arXiv2022

Problems and results on 1-cross intersecting set pair systems

The notion of cross intersecting set pair system of size $m$, $\Big(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m\Big)$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne\emptyset$, was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le {a+b\choose a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$. Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is -- at least $5^{n/2}$ for $n$ even, $a=b=n$, -- equal to $\bigl(\lfloor\frac{n}{2}\rfloor+1\bigr)\bigl(\lceil\frac{n}{2}\rceil+1\bigr)$ if $a=2$ and $b=n\ge 4$, -- at most $|\cup_{i=1}^m A_i|$, -- asymptotically $n^2$ if $\{A_i\}$ is a linear hypergraph ($|A_i\cap A_j|\le 1$ for $i\ne j$), -- asymptotically ${1\over 2}n^2$ if $\{A_i\}$ and $\{B_i\}$ are both linear hypergraphs.

math.CO↗

arXiv2021

How many matchings cover the nodes of a graph?

Given an undirected graph, are there $k$ matchings whose union covers all of its nodes, that is, a matching-$k$-cover? A first, easy polynomial solution from matroid union is possible, as already observed by Wang, Song and Yuan (Mathematical Programming, 2014). However, it was not satisfactory neither from the algorithmic viewpoint nor for proving graphic theorems, since the corresponding matroid ignores the edges of the graph. We prove here, simply and algorithmically: all nodes of a graph can be covered with $k\ge 2$ matchings if and only if for every stable set $S$ we have $|S|\le k\cdot|N(S)|$. When $k=1$, an exception occurs: this condition is not enough to guarantee the existence of a matching-$1$-cover, that is, the existence of a perfect matching, in this case Tutte's famous matching theorem (J. London Math. Soc., 1947) provides the right `good' characterization. The condition above then guarantees only that a perfect $2$-matching exists, as known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953). Some results are then deduced as consequences with surprisingly simple proofs, using only the level of difficulty of bipartite matchings. We give some generalizations, as well as a solution for minimization if the edge-weights are non-negative, while the edge-cardinality maximization of matching-$2$-covers turns out to be already NP-hard. We have arrived at this problem as the line graph special case of a model arising for manufacturing integrated circuits with the technology called `Directed Self Assembly'.

math.CO↗

arXiv2020

Upper bounds for the necklace folding problems

A necklace can be considered as a cyclic list of $n$ red and $n$ blue beads in an arbitrary order, and the goal is to fold it into two and find a large cross-free matching of pairs of beads of different colors. We give a counterexample for a conjecture about the necklace folding problem, also known as the separated matching problem. The conjecture (given independently by three sets of authors) states that $μ=\frac{2}{3}$, where $μ$ is the ratio of the `covered' beads to the total number of beads. We refute this conjecture by giving a construction which proves that $μ\le 2 \nolinebreak - \nolinebreak \sqrt 2 < 0.5858$. Our construction also applies to the homogeneous model: when we are matching beads of the same color. Moreover, we also consider the problem where the two color classes not necessarily have the same size.

math.CO↗

arXiv2018

The Manickam-Miklós-Singhi Parameter of Graphs and Degree Sequences

Let $G$ be a simple graph. Consider all weightings of the vertices of $G$ with real numbers whose total sum is nonnegative. How many edges of $G$ have endpoints with a nonnegative sum? We consider the minimum number of such edges over all such weightings as a graph parameter. Computing this parameter has been shown to be NP-hard but we give a polynomial algorithm to compute the minimum of this parameter over realizations of a given degree sequence. We also completely determine the minimum and maximum value of this parameter for regular graphs.

math.CO↗

arXiv2018

Acyclic orientations with degree constraints

In this note we study the complexity of some generalizations of the notion of $st$-numbering. Suppose that given some functions $f$ and $g$, we want to order the vertices of a graph such that every vertex $v$ is preceded by at least $f(v)$ of its neighbors and succeeded by at least $g(v)$ of its neighbors. We prove that this problem is solvable in polynomial time if $fg\equiv 0$, but it becomes NP-complete for $f\equiv g \equiv 2$. This answers a question of the first author posed in 2009.

cs.CC↗

arXiv2017

Network Coding Algorithms for Multi-Layered Video Broadcast

In this paper we give network coding algorithms for multi-layered video streaming. The problem is motivated by video broadcasting in a communication network to users with varying demands. We give a polynomial time algorithm for deciding feasibility for the case of two layers, and show that the problem becomes NP-hard if the task is to maximize the number of satisfied demands. For the case of three layers we also show NP-hardness of the problem. Finally, we propose a heuristic for three layers and give experimental comparison with previous approaches.

cs.DM↗

arXiv2017

Spanning tree with lower bound on the degrees

We concentrate on some recent results of Egawa and Ozeki [J. Graph Theory, 2015 and Combinatorica, 2014], and He et al. [J. Graph Theory, 2002]. We give shorter proofs and polynomial time algorithms as well. We present two new proofs for the sufficient condition for having a spanning tree with prescribed lower bounds on the degrees, achieved recently by Egawa and Ozeki. The first one is a natural proof using induction, and the second one is a simple reduction to the theorem of Lovász. Using an algorithm of Frank we show that the condition of the theorem can be checked in time $O(m\sqrt{n})$, and moreover, in the same running time -- if the condition is satisfied -- we can also generate the spanning tree required. This gives the first polynomial time algorithm for this problem. Next we show a nice application of this theorem for the simplest case of the Weak Nine Dragon Tree Conjecture, and for the game coloring number of planar graphs, first discovered by He et al. Finally, we give a shorter proof and a polynomial time algorithm for a good characterization of having a spanning tree with prescribed degree lower bounds, for the special case when $G[S]$ is a cograph, where $S$ is the set of the vertices having degree lower bound prescription at least two. This theorem was proved by Egawa and Ozeki in 2014 while they did not give a polynomial time algorithm.

math.CO↗

arXiv2017

Packing tree degree sequences

We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. This theorem can be extended to more trees: we show that the edge packing problem is in P for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees. We also consider the following variant of the degree matrix problem: given two degree sequences $D_1$ and $D_2$ such that $D_2$ is a tree degree sequence, decide if there exists edge-disjoint realizations of $D_1$ and $D_2$ where the realization of $D_2$ is not necessarily a tree. We show that this problem is already $\NP$-complete. Counting, or just estimating the number of distinct realizations of degree sequences is challenging in general. We show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.

math.CO↗

arXiv2016

Covering complete partite hypergraphs by monochromatic components

A well-known special case of a conjecture attributed to Ryser states that k-partite intersecting hypergraphs have transversals of at most k-1 vertices. An equivalent form was formulated by Gyárfás: if the edges of a complete graph K are colored with k colors then the vertex set of K can be covered by at most k-1 sets, each connected in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: Z. Király proved that in every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >= 3), the vertex set of K^r can be covered by at most $\lceil k/r \rceil$ sets, each connected in some color. Here we investigate the analogue problem for complete r-uniform r-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of any color used in the coloring. We propose the following analogue of Ryser conjecture. In every spanning (r+t)-coloring of the edges of a complete r-uniform r-partite hypergraph, the vertex set can be covered by at most t+1 sets, each connected in some color. Our main result is that the conjecture is true for 1 <= t <= r-1. We also prove a slightly weaker result for t >= r, namely that t+2 sets, each connected in some color, are enough to cover the vertex set. To build a bridge between complete r-uniform and complete r-uniform r-partite hypergraphs, we introduce a new notion. A hypergraph is complete r-uniform (r,l)-partite if it has all r-sets that intersect each partite class in at most l vertices. Extending our results achieved for l=1, we prove that for any r >= 3, 2 <= l <= r, k >= 1+r-l, in every spanning k-coloring of the edges of a complete r-uniform (r,l)-partite hypergraph, the vertex set can be covered by at most 1+\lfloor \frac{k-r+\ell-1}{\ell}\rfloor sets, each connected in some color.

math.CO↗

arXiv2014

Notes on dual-critical graphs

We define dual-critical graphs as graphs having an acyclic orientation, where the indegrees are odd except for the unique source. We have very limited knowledge about the complexity of dual-criticality testing. By the definition the problem is in NP, and a result of Balázs and Christian Szegedy provides a randomized polynomial algorithm, which relies on formal matrix rank computing. It is unknown whether dual-criticality test can be done in deterministic polynomial time. Moreover, the question of being in co-NP is also open. We give equivalent descriptions for dual-critical graphs in the general case, and further equivalent descriptions in the special cases of planar graphs and 3-regular graphs. These descriptions provide polynomial algorithms for these special classes. We also give an FPT algorithm for a relaxed version of dual-criticality called $k$-dual-criticality.

cs.DS↗

arXiv2014

Shortest Paths in Nearly Conservative Digraphs

We introduce the following notion: a digraph $D=(V,A)$ with arc weights $c: A\rightarrow \R$ is called nearly conservative if every negative cycle consists of two arcs. Computing shortest paths in nearly conservative digraphs is NP-hard, and even deciding whether a digraph is nearly conservative is coNP-complete. We show that the "All Pairs Shortest Path" problem is fixed parameter tractable with various parameters for nearly conservative digraphs. The results also apply for the special case of conservative mixed graphs.

cs.DS↗

arXiv2014

On the tractability of some natural packing, covering and partitioning problems

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mathrm{B}$? \textbf{Covering problem:} can we cover $E$ with an object of type $\mathrm{A}$, and an object of type $\mathrm{B}$? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an $s$-$t$ path $P$ and an $s'$-$t'$ path $P'$ that are edge-disjoint. However, many others were not, for example can we find an $s$-$t$ path $P\subseteq E $ and a spanning tree $T\subseteq E$ that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense).

cs.CC↗

arXiv2013

On the swap-distances of different realizations of a graphical degree sequence

One of the first graph theoretical problems which got serious attention (already in the fifties of the last century) was to decide whether a given integer sequence is equal to the degree sequence of a simple graph (or it is {\em graphical} for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on the {\em swap} operation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another one of the same degree sequence. This latter problem got particular emphases in connection of fast mixing Markov chain approaches to sample uniformly all possible realizations of a given degree sequence. (This becomes a matter of interest in connection of -- among others -- the study of large social networks.) Earlier there were only crude upper bounds on the shortest possible length of such swap sequences between two realizations. In this paper we develop formulae (Gallai-type identities) for these {\em swap-distance}s of any two realizations of simple undirected or directed degree sequences. These identities improves considerably the known upper bounds on the swap-distances.

math.CO↗