arXiv subjects

Máté Vizer

Publications and source records attributed to Máté Vizer.

At least 19 recordsLinked to original sources

arXiv2026

Edge mappings of graphs: Ramsey type parameters

In this paper, we launch a systematic study of parameters concerning edge mappings of graphs. Inspired by Ramsey's theorem, the quantity $m(G,H)$ is defined to be the smallest integer $n$ such that for every $f:E(K_n)\rightarrow E(K_n)$ either there is an $f$-fixed copy of $G$ with $f(e)=e$ for all $e\in E(G)$, or an $f$-free copy of $H$ with $f(e)\notin E(H)$ for all $e\in E(H)$. Incorporating new ideas, we extend many old results from the 1980s and prove many new exact results, mostly concerning $m(T,K_r)$, where $T$ is a tree, and $m(G,K_{1,r})$. We also study further related parameters, most of them introduced in the 1980s, and obtain substantial progress regarding these parameters.

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arXiv2025

Generalized saturation game

We study the following game version of the generalized graph Turán problem. For two fixed graphs F and H, two players, Max and Mini, alternately claim unclaimed edges of the complete graph Kn such that the graph G of the claimed edges must remain F-free throughout the game. The game ends when no further edges can be claimed, i.e. when G becomes F-saturated. The H-score of the game is the number of copies of H in G. Max aims to maximize the H-score, while Mini wants to minimize it. The H-score of the game when both players play optimally is denoted by s1(n, #H, F) when Max starts, and by s2(n, #H, F) when Mini starts. We study these values for several natural choices of F and H.

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arXiv2025

The generalized trifference problem

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.

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arXiv2024

Edge mappings of graphs: Turán type parameters

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ for all $e\in E$ and further in all copies $G'$ of $G$ in $H$ there exists $e\in E(G')$ with $f(e)\in E(G')$. Among other results, we determine $h(n, G)$ when $G$ is a matching and $n$ is large enough. As a related concept, we say that $H$ is unavoidable for $G$ if for any mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ there exists a copy $G'$ of $G$ in $H$ such that $f(e)\notin E(G')$ for all $e\in E(G)$. The set of minimal unavoidable graphs for $G$ is denoted by $\mathcal{M}(G)$. We prove that if $F$ is a forest, then $\mathcal{M}(F)$ is finite if and only if $F$ is a matching, and we conjecture that for all non-forest graphs $G$, the set $\mathcal{M}(G)$ is infinite. Several other parameters are defined with basic results proved. Lots of open problems remain.

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arXiv2023

Counting Connected Partitions of Graphs

Motivated by the theorem of Gy\H ori and Lovász, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$ such that every $m_i$ is realized by a connected subgraph $H_i$ of $G$ with $m_i$ edges such that $\cup_{i=1}^kE(H_i)=E(G)$. We also consider the vertex-partition analogue. We prove various lower bounds on $P(G,k)$ as a function of the number $n$ of vertices in $G$, as a function of the average degree $d$ of $G$, and also as the size $\mathrm{CMC}_r(G)$ of $r$-partite connected maximum cuts of $G$. Those three lower bounds are tight up to a multiplicative constant. We also prove that the number $π(G,k)$ of unordered $k$-tuples with $\sum_{i=1}^kn_i=n$, that are realizable by vertex partitions into $k$ connected parts of respective sizes $n_1,n_2,\dots,n_k$, is $Ω(d^{k-1})$.

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arXiv2023

The Constructor-Blocker Game

We study the following game version of the generalized graph Turán problem. For two fixed graphs $F$ and $H$, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph $K_n$. Constructor can only claim edges so that he never claims all edges of any copy of $F$, i.e. his graph must remain $F$-free, while Blocker can claim unclaimed edges without restrictions. The game ends when Constructor cannot claim further edges or when all edges have been claimed. The score of the game is the number of copies of $H$ with all edges claimed by Constructor. Constructor's aim is to maximize the score, while Blocker tries to keep the score as low as possible. We denote by $g(n,H,F)$ the score of the game when both players play optimally and Constructor starts the game. In this paper, we obtain the exact value of $g(n,H,F)$ when both $F$ and $H$ are stars and when $F=P_4$, $H=P_3$. We determine the asymptotics of $g(n,H,F)$ when $F$ is a star and $H$ is a tree and when $F=P_5$, $H=K_3$, and we derive upper and lower bounds on $g(n,P_4,P_5)$.

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arXiv2023

Stability of extremal connected hypergraphs avoiding Berge-paths

A Berge-path of length $k$ in a hypergraph $\mathcal{H}$ is a sequence $v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_{i},v_{i+1} \in e_i$, for $i \le k$. Füredi, Kostochka and Luo, and independently Győri, Salia and Zamora determined the maximum number of hyperedges in an $n$-vertex, connected, $r$-uniform hypergraph that does not contain a Berge-path of length $k$ provided $k$ is large enough compared to $r$. They also determined the unique extremal hypergraph $\mathcal{H}_1$. We prove a stability version of this result by presenting another construction $\mathcal{H}_2$ and showing that any $n$-vertex, connected, $r$-uniform hypergraph without a Berge-path of length $k$, that contains more than $|\mathcal{H}_2|$ hyperedges must be a sub-hypergraph of the extremal hypergraph $\mathcal{H}_1$, provided $k$ is large enough compared to $r$.

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arXiv2023

Extremal graph theoretic questions for q-ary vectors

A $q$-graph $H$ on $n$ vertices is a set of vectors of length $n$ with all entries from $\{0,1,\dots,q\}$ and every vector (that we call a $q$-edge) having exactly two non-zero entries. The support of a $q$-edge $\mathbf{x}$ is the pair $S_{\mathbf{x}}$ of indices of non-zero entries. We say that $H$ is an $s$-copy of an ordinary graph $F$ if $|H|=|E(F)|$, $F$ is isomorphic to the graph with edge set $\{S_{\mathbf{x}}:\mathbf{x}\in H\}$, and whenever $v\in e,e'\in E(F)$, the entries with index corresponding to $v$ in the $q$-edges corresponding to $e$ and $e'$ sum up to at least $s$. E.g., the $q$-edges $(1,3,0,0,0), (0,1,0,0,3)$, and $(3,0,0,0,1)$ form a 4-triangle. The Turán number $\mathrm{ex}(n,F,q,s)$ is the maximum number of $q$-edges that a $q$-graph $H$ on $n$ vertices can have if it does not contain any $s$-copies of $F$. In the present paper, we determine the asymptotics of $\mathrm{ex}(n,F,q,q+1)$ for many graphs $F$.

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arXiv2023

The robust chromatic number of graphs

A 1-removed subgraph $G_f$ of a graph $G=(V,E)$ is obtained by $(i)$ selecting at most one edge $f(v)$ for each vertex $v\in V$, such that $v\in f(v)\in E$ (the mapping $f:V\to E \cup \{\varnothing\}$ is allowed to be non-injective), and $(ii)$ deleting all the selected edges $f(v)$ from the edge set $E$ of $G$. Proper vertex colorings of 1-removed subgraphs proved to be a useful tool for earlier research on some Turán-type problems. In this paper, we introduce a systematic investigation of the graph invariant 1-robust chromatic number, denoted as $ω_1(G)$. This invariant is defined as the minimum chromatic number $χ(G_f)$ among all 1-removed subgraphs $G_f$ of $G$. We also examine other standard graph invariants in a similar manner.

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arXiv2023

The robust chromatic number of certain graph classes

A 1-selection $f$ of a graph $G$ is a function $f: V(G)\rightarrow E(G)$ such that $f(v)$ is incident to $v$ for every vertex $v$. The 1-removed $G_f$ is the graph $(V(G),E(G)\setminus f[V(G)])$. The (1-)robust chromatic number $χ_1(G)$ is the minimum of $χ(G_f)$ over all 1-selections $f$ of $G$. We determine the robust chromatic number of complete multipartite graphs and Kneser graphs and prove tight lower and upper bounds on the robust chromatic number of chordal graphs and some of their extensively studied subclasses, with respect to their ordinary chromatic number.

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arXiv2023

Vector sum-intersection theorems

We introduce the following generalization of set intersection via characteristic vectors: for $n,q,s, t \ge 1$ a family $\mathcal{F}\subseteq \{0,1,\dots,q\}^n$ of vectors is said to be \emph{$s$-sum $t$-intersecting} if for any distinct $\mathbf{x},\mathbf{y}\in \mathcal{F}$ there exist at least $t$ coordinates, where the entries of $\mathbf{x}$ and $\mathbf{y}$ sum up to at least $s$, i.e.\ $|\{i:x_i+y_i\ge s\}|\ge t$. The original set intersection corresponds to the case $q=1,s=2$. We address analogs of several variants of classical results in this setting: the Erdős--Ko--Rado theorem and the theorem of Bollobás on intersecting set pairs.

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arXiv2022

On ordered Ramsey numbers of tripartite 3-uniform hypergraphs

For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$ of two ordered $k$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{G}$ is the smallest $N \in \mathbb{N}$ such that every red-blue coloring of the hyperedges of the ordered complete $k$-uniform hypergraph $\mathcal{K}^{(k)}_N$ on $N$ vertices contains a blue copy of $\mathcal{H}$ or a red copy of $\mathcal{G}$. The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all $d,n \in \mathbb{N}$ and for every ordered $3$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with maximum degree $d$ and with interval chromatic number $3$ there is an $\varepsilon=\varepsilon(d)>0$ such that $$\overline{R}(\mathcal{H},\mathcal{H}) \leq 2^{O(n^{2-\varepsilon})}.$$ In fact, we prove this upper bound for the number $\overline{R}(\mathcal{G},\mathcal{K}^{(3)}_3(n))$, where $\mathcal{G}$ is an ordered 3-uniform hypergraph with $n$ vertices and maximum degree $d$ and $\mathcal{K}^{(3)}_3(n)$ is the ordered complete tripartite hypergraph with consecutive color classes of size $n$. We show that this bound is not far from the truth by proving $\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n)) \geq 2^{Ω(n\log{n})}$ for some fixed ordered $3$-uniform hypergraph $\mathcal{H}$.

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arXiv2022

On graphs that contain exactly k copies of a subgraph, and a related problem in search theory

We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Turán number $\mathrm{ex}(n,F)$ that is among the most studied parameters in extremal graph theory. We show that for any $F$ and $k$, $\mathrm{exa}_k(n,F)=(1+o(1))\mathrm{ex}(n,F))$ and determine the exact values of $\mathrm{exa}_k(n,K_3)$ and $\mathrm{exa}_1(n,K_r)$ for $n$ large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order $n$ that consists of an unknown copy of $F$ and some isolated vertices. We can ask pairs of vertices as queries, and the answer tells us whether there is an edge between those vertices. Our goal is to describe the graph using as few queries as possible. Aigner and Triesch in 1990 showed that the number of queries needed is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$. Among other results we show that the number of queries that were answered NO is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$.

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arXiv2021

On non-adaptive majority problems of large query size

We are given $n$ balls and an unknown coloring of them with two colors. Our goal is to find a ball that belongs to the larger color class, or show that the color classes have the same size. We can ask sets of $k$ balls as queries, and the problem has different variants, according to what the answers to the queries can be. These questions has attracted several researchers, but the focus of most research was the adaptive version, where queries are decided sequentially, after learning the answer to the previous query. Here we study the non-adaptive version, where all the queries have to be asked at the same time.

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arXiv2021

Forbidden subposet problems in the grid

For posets $P$ and $Q$, extremal and saturation problems about weak and strong $P$-free subposets of $Q$ have been studied mostly in the case $Q$ is the Boolean poset $Q_n$, the poset of all subsets of an $n$-element set ordered by inclusion. In this paper, we study some instances of the problem with $Q$ being the grid, and its connections to the Boolean case and to the forbidden submatrix problem.

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arXiv2021

Turán problems for Edge-ordered graphs

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the edge-order. A subgraph of an edge-ordered graph is itself an edge-ordered graph with the induced edge-order. We say that an edge-ordered graph $G$ $\textit{avoids}$ another edge-ordered graph $H$, if no subgraph of $G$ is isomorphic to $H$. The $\textit{Turán number}$ of an edge-ordered graph $H$ is the maximum number of edges in an edge-ordered graph on $n$ vertices that avoids $H$. We study this problem in general, and establish an Erdős-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Turán number of an edge-ordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Turán numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in Discrete Geometry.

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arXiv2021

Unified approach to the generalized Turán problem and supersaturation

In this paper we introduce a unifying approach to the generalized Turán problem and supersaturation results in graph theory. The supersaturation-extremal function $satex(n, F : m, G)$ is the least number of copies of a subgraph $G$ an $n$-vertex graph can have, which contains at least $m$ copies of $F$ as a subgraph. We present a survey, discuss previously known results and obtain several new ones focusing mainly on proof methods, extremal structure and phase transition phenomena. Finally we point out some relation with extremal questions concerning hypergraphs, particularly Berge-type results.

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arXiv2021

Edge-ordered Ramsey numbers

We introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey number $\overline{R}_e(\mathfrak{G})$ of an edge-ordered graph $\mathfrak{G}$ is the minimum positive integer $N$ such that there exists an edge-ordered complete graph $\mathfrak{K}_N$ on $N$ vertices such that every 2-coloring of the edges of $\mathfrak{K}_N$ contains a monochromatic copy of $\mathfrak{G}$ as an edge-ordered subgraph of $\mathfrak{K}_N$. We prove that the edge-ordered Ramsey number $\overline{R}_e(\mathfrak{G})$ is finite for every edge-ordered graph $\mathfrak{G}$ and we obtain better estimates for special classes of edge-ordered graphs. In particular, we prove $\overline{R}_e(\mathfrak{G}) \leq 2^{O(n^3\log{n})}$ for every bipartite edge-ordered graph $\mathfrak{G}$ on $n$ vertices. We also introduce a natural class of edge-orderings, called lexicographic edge-orderings, for which we can prove much better upper bounds on the corresponding edge-ordered Ramsey numbers.

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