We propose a new approach for defining and searching clusters in graphs that represent real techn... more We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied.
In their previous work, the authors considered the concept of the random spanning tree intersecti... more In their previous work, the authors considered the concept of the random spanning tree intersection value of complex networks []. A simple formula was derived for the minimum expected intersection value of two spanning trees chosen uniformly at random, and Monte Carlo experiments were run for real networks. In this paper, we provide a broader context and motivation for the concept, discussing its game theoretic origins, examples, its applications in network optimization problems, and its potential use in quantifying the resilience and modular structure of complex networks.
In the literature, the notion of discrepancy is used in several contexts, even in the theory of g... more In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph G, {−1, 1} labels are assigned to the edges, and we consider a family S G of (spanning) subgraphs of certain types, among others spanning trees, Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of S G , for every labeling.
In this paper, we investigate special types of Maker-Breaker games defined on graphs. We restrict... more In this paper, we investigate special types of Maker-Breaker games defined on graphs. We restrict Maker's possible moves that resembles the way that was introduced by Espig, Frieze, Krivelevich and Pedgen [9]. Here, we require that the subgraph induced by Maker's edges must be connected throughout the game. Besides the normal play, we examine the biased and accelerated versions of these games.
We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied f... more We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied for a large class of graphs. The applications involve random graphs and a weighted version of the Erdős-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set.
We dene a modied PageRank algorithm and the P R-score to measure the inuence of a single article ... more We dene a modied PageRank algorithm and the P R-score to measure the inuence of a single article by using its local co-citation network. We also calculate the reaching probability and RP-score of a paper starting at an arbitrary article of its co-citation network for the same purpose. We highlight the advantages of our methods by applying them on the celebrated paper of Jen® Egerváry that is underrated by the standard indices.
HAL (Le Centre pour la Communication Scientifique Directe), 2004
The interval number of a graph G is the least natural number t such that G is the intersection gr... more The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i (G). Griggs and West showed that i(G) d 1 2 (d þ 1)e. We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independence number. Finally, we show that a graph needs to contain large complete bipartite induced subgraphs in order to have interval number larger than the random graph on the same number of vertices.
A pozíciós játékok többnyire véges, kétszemélyes, zérusösszegű, teljes információs játékok, amely... more A pozíciós játékok többnyire véges, kétszemélyes, zérusösszegű, teljes információs játékok, amelyekben még kevert stratégiák alkalmazására sincs szükség. A pozíciós játékok megadási módjuk miatt viszont mátrixjátékként nem kezelhetők jól, így a vizsgálatukra változatos matematikai eszköztár alakult ki. Jelen áttekintés célja ezek vázlatos ismertetése, néhol kiterjesztése
Régóta ismert a lineáris algebraés a kombinatorika kapcsolata. Itt a Kronecker-Capelli tétel komb... more Régóta ismert a lineáris algebraés a kombinatorika kapcsolata. Itt a Kronecker-Capelli tétel kombinatorikai következményeit járjuk körbe, mely kiadja Kőnigés Harary tételeités elvezet egyfajta duálisaikhoz, valamint a nyaklánc problémaáltalánosításaihozés specializációihoz.
We propose a new approach for defining and searching clusters in graphs that represent real techn... more We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied.
A path separator of a graph G is a set of paths P = {P1, . . . , Pt} such that for every pair of ... more A path separator of a graph G is a set of paths P = {P1, . . . , Pt} such that for every pair of edges e, f ∈ E(G), there exist paths Pe, Pf ∈ P such that e ∈ E(Pe), f 6∈ E(Pe), e 6∈ E(Pf ) and f ∈ E(Pf ). The path separation number of G, denoted psn(G), is the smallest number of paths in a path separator. We shall estimate the path separation number of several graph families – including complete graphs, random graph, the hypercube – and discuss general graphs as well.
A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G ... more A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-tiling in a graph G. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks substructures F of G that have ‘high’ discrepancy (i.e. the sum of the labels of the edges in F is far from 0). In this paper we determine the minimum degree threshold for a graph to contain a perfect Kr-tiling of high discrepancy.
A local food wholesaler company is using an automated commissioning system, which brings the bins... more A local food wholesaler company is using an automated commissioning system, which brings the bins containing the appropriate product to the commissioning counter, where the worker picks the needed amounts to 12 bins corresponding to the same number of orders. To minimize the number of bins to pick from, they pick for several different spreading tours, so the order of bins containing the picked products coming from the commissioning counter can be considered random in this sense. Recently, the number of bins containing the picked orders increased over the available storage space, and it was necessary to find a new way of storing and ordering the bins to spreading tours. We developed a conveyor system which (after a preprocessing step) can order the bins in linear space and time.
In the literature, the notion of discrepancy is used in several contexts, even in the theory of g... more In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$ with each edge labelled $-1$ or $1$, we consider a family $\mathcal{S}_G$ of subgraphs of a certain type, such as spanning trees or Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of $\mathcal{S}_G$, for every labeling.
In this paper we define a bridge between pairings and colorings of the hypergraphs by introducing... more In this paper we define a bridge between pairings and colorings of the hypergraphs by introducing a generalization of pairs called t-cakes for t ∈ ℕ, t ≥ 2. For t = 2 the 2-cakes are the same as the well-known pairs of system of distinct representatives, that can be turned to pairing strategies in Maker-Breaker hypergraph games, see Hales and JewettThe method also gives bounds on the condition of winnings in certain biased Chooser-Picker games, which can be introduced similarly to Beck
Several methods have been proposed recently to estimate the edge infection probabilities in infec... more Several methods have been proposed recently to estimate the edge infection probabilities in infection or diffusion models. In this paper we will use the framework of the Generalized Cascade Model to define the Inverse Infection Problem-the problem of calculating these probabilities. We are going to show that the problem can be reduced to an optimization task and we will give a particle swarm based method as a solution. We will show, that direct estimation of the separate edge infection values is possible, although only on small graphs with a few thousand edges. To reduce the dimensionality of the task, the edge infection values can be considered as functions of known attributes on the vertices or edges of the graph, this way only the unknown coefficients of these functions have to be estimated. We are going to evaluate our method on artificially created infection scenarios. Our main points of interest are the accuracy and stability of the estimation.
One of the most useful strategies for proving Breaker's win in Maker-Breaker Positional Games is ... more One of the most useful strategies for proving Breaker's win in Maker-Breaker Positional Games is to find a pairing strategy. In some cases there are no pairing strategies at all, in some cases there are unique or almost unique strategies. For the kin a row game, the case k = 9 is the smallest (sharp) for which there exists a Breaker winning pairing (paving) strategy. One pairing strategy for this game was given by Hales and Jewett. In this paper we show that there are other winning pairings for the 9-in-a-row game, all have a very symmetric torus structure. While describing these symmetries we prove that there are only a finite number of non-isomorphic pairings for the game (around 200 thousand), which can be also listed up by a computer program. In addition, we prove that there are no "irregular", non-symmetric pairings. At the end of the paper we also show a pairing strategy for a variant of the 3-dimensional kin a row game.
We propose a new approach for defining and searching clusters in graphs that represent real techn... more We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied.
In their previous work, the authors considered the concept of the random spanning tree intersecti... more In their previous work, the authors considered the concept of the random spanning tree intersection value of complex networks []. A simple formula was derived for the minimum expected intersection value of two spanning trees chosen uniformly at random, and Monte Carlo experiments were run for real networks. In this paper, we provide a broader context and motivation for the concept, discussing its game theoretic origins, examples, its applications in network optimization problems, and its potential use in quantifying the resilience and modular structure of complex networks.
In the literature, the notion of discrepancy is used in several contexts, even in the theory of g... more In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph G, {−1, 1} labels are assigned to the edges, and we consider a family S G of (spanning) subgraphs of certain types, among others spanning trees, Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of S G , for every labeling.
In this paper, we investigate special types of Maker-Breaker games defined on graphs. We restrict... more In this paper, we investigate special types of Maker-Breaker games defined on graphs. We restrict Maker's possible moves that resembles the way that was introduced by Espig, Frieze, Krivelevich and Pedgen [9]. Here, we require that the subgraph induced by Maker's edges must be connected throughout the game. Besides the normal play, we examine the biased and accelerated versions of these games.
We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied f... more We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied for a large class of graphs. The applications involve random graphs and a weighted version of the Erdős-Stone theorem. We also provide means to handle the otherwise uncontrolled exceptional set.
We dene a modied PageRank algorithm and the P R-score to measure the inuence of a single article ... more We dene a modied PageRank algorithm and the P R-score to measure the inuence of a single article by using its local co-citation network. We also calculate the reaching probability and RP-score of a paper starting at an arbitrary article of its co-citation network for the same purpose. We highlight the advantages of our methods by applying them on the celebrated paper of Jen® Egerváry that is underrated by the standard indices.
HAL (Le Centre pour la Communication Scientifique Directe), 2004
The interval number of a graph G is the least natural number t such that G is the intersection gr... more The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i (G). Griggs and West showed that i(G) d 1 2 (d þ 1)e. We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independence number. Finally, we show that a graph needs to contain large complete bipartite induced subgraphs in order to have interval number larger than the random graph on the same number of vertices.
A pozíciós játékok többnyire véges, kétszemélyes, zérusösszegű, teljes információs játékok, amely... more A pozíciós játékok többnyire véges, kétszemélyes, zérusösszegű, teljes információs játékok, amelyekben még kevert stratégiák alkalmazására sincs szükség. A pozíciós játékok megadási módjuk miatt viszont mátrixjátékként nem kezelhetők jól, így a vizsgálatukra változatos matematikai eszköztár alakult ki. Jelen áttekintés célja ezek vázlatos ismertetése, néhol kiterjesztése
Régóta ismert a lineáris algebraés a kombinatorika kapcsolata. Itt a Kronecker-Capelli tétel komb... more Régóta ismert a lineáris algebraés a kombinatorika kapcsolata. Itt a Kronecker-Capelli tétel kombinatorikai következményeit járjuk körbe, mely kiadja Kőnigés Harary tételeités elvezet egyfajta duálisaikhoz, valamint a nyaklánc problémaáltalánosításaihozés specializációihoz.
We propose a new approach for defining and searching clusters in graphs that represent real techn... more We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied.
A path separator of a graph G is a set of paths P = {P1, . . . , Pt} such that for every pair of ... more A path separator of a graph G is a set of paths P = {P1, . . . , Pt} such that for every pair of edges e, f ∈ E(G), there exist paths Pe, Pf ∈ P such that e ∈ E(Pe), f 6∈ E(Pe), e 6∈ E(Pf ) and f ∈ E(Pf ). The path separation number of G, denoted psn(G), is the smallest number of paths in a path separator. We shall estimate the path separation number of several graph families – including complete graphs, random graph, the hypercube – and discuss general graphs as well.
A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G ... more A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-tiling in a graph G. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks substructures F of G that have ‘high’ discrepancy (i.e. the sum of the labels of the edges in F is far from 0). In this paper we determine the minimum degree threshold for a graph to contain a perfect Kr-tiling of high discrepancy.
A local food wholesaler company is using an automated commissioning system, which brings the bins... more A local food wholesaler company is using an automated commissioning system, which brings the bins containing the appropriate product to the commissioning counter, where the worker picks the needed amounts to 12 bins corresponding to the same number of orders. To minimize the number of bins to pick from, they pick for several different spreading tours, so the order of bins containing the picked products coming from the commissioning counter can be considered random in this sense. Recently, the number of bins containing the picked orders increased over the available storage space, and it was necessary to find a new way of storing and ordering the bins to spreading tours. We developed a conveyor system which (after a preprocessing step) can order the bins in linear space and time.
In the literature, the notion of discrepancy is used in several contexts, even in the theory of g... more In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$ with each edge labelled $-1$ or $1$, we consider a family $\mathcal{S}_G$ of subgraphs of a certain type, such as spanning trees or Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of $\mathcal{S}_G$, for every labeling.
In this paper we define a bridge between pairings and colorings of the hypergraphs by introducing... more In this paper we define a bridge between pairings and colorings of the hypergraphs by introducing a generalization of pairs called t-cakes for t ∈ ℕ, t ≥ 2. For t = 2 the 2-cakes are the same as the well-known pairs of system of distinct representatives, that can be turned to pairing strategies in Maker-Breaker hypergraph games, see Hales and JewettThe method also gives bounds on the condition of winnings in certain biased Chooser-Picker games, which can be introduced similarly to Beck
Several methods have been proposed recently to estimate the edge infection probabilities in infec... more Several methods have been proposed recently to estimate the edge infection probabilities in infection or diffusion models. In this paper we will use the framework of the Generalized Cascade Model to define the Inverse Infection Problem-the problem of calculating these probabilities. We are going to show that the problem can be reduced to an optimization task and we will give a particle swarm based method as a solution. We will show, that direct estimation of the separate edge infection values is possible, although only on small graphs with a few thousand edges. To reduce the dimensionality of the task, the edge infection values can be considered as functions of known attributes on the vertices or edges of the graph, this way only the unknown coefficients of these functions have to be estimated. We are going to evaluate our method on artificially created infection scenarios. Our main points of interest are the accuracy and stability of the estimation.
One of the most useful strategies for proving Breaker's win in Maker-Breaker Positional Games is ... more One of the most useful strategies for proving Breaker's win in Maker-Breaker Positional Games is to find a pairing strategy. In some cases there are no pairing strategies at all, in some cases there are unique or almost unique strategies. For the kin a row game, the case k = 9 is the smallest (sharp) for which there exists a Breaker winning pairing (paving) strategy. One pairing strategy for this game was given by Hales and Jewett. In this paper we show that there are other winning pairings for the 9-in-a-row game, all have a very symmetric torus structure. While describing these symmetries we prove that there are only a finite number of non-isomorphic pairings for the game (around 200 thousand), which can be also listed up by a computer program. In addition, we prove that there are no "irregular", non-symmetric pairings. At the end of the paper we also show a pairing strategy for a variant of the 3-dimensional kin a row game.
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