arXiv subjects

Ervin Győri

Publications and source records attributed to Ervin Győri.

At least 37 records · Page 2Linked to original sources

arXiv2023

A new construction for planar Turán number of cycle

The planar Turán number ${\rm ex}_{\mathcal{P}}(n,C_k)$ is the largest number of edges in an $n$-vertex planar graph with no cycle of length $k$. Let $k\ge 11$ and $C,D$ be constants. Cranston, Lidický, Liu and Shantanam \cite{2021Planar}, and independently Lan and Song \cite{LanSong} showed that ${\rm ex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{Cn}{k}$ for large $n$. Moreover, Cranston et al. conjectured that ${\rm ex}_{\mathcal{P}}(n,C_k)\le 3n-6-\frac{Dn}{k^{lg_23}}$ when $n$ is large. In this note, we prove that ${\rm ex}_{\mathcal{P}}(n,C_k)\ge 3n-6-\frac{6\cdot 3^{lg_23}n}{k^{lg_23}}$ for every $k$. It implies Cranston et al.'s conjecture is essentially best possible.

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arXiv2023

Outerplanar Turán number of a cycle

A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let $H$ be a graph. The outerplanar Turán number of $H$, denoted by $ex_\mathcal{OP}(n,H)$, is the maximum number of edges in an $n$-vertex outerplanar graph which does not contain $H$ as a subgraph. In 2021, L. Fang et al. determined the outerplanar Turán number of cycles and paths. In this paper, we use techniques of dual graph to give a shorter proof for the sharp upperbound of $ex_\mathcal{OP}(n,C_k)\leq \frac{(2k - 5)(kn - k - 1)}{k^2 - 2k - 1}$.

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arXiv2023

Planar Turán Number of the $Θ_6$

Let $\mathcal{F}$ be a nonempty family of graphs. A graph $G$ is called $\mathcal{F}$-\textit{free} if it contains no graph from $\mathcal{F}$ as a subgraph. For a positive integer $n$, the \emph{planar Turán number} of $\F$, denoted by $\ex_{\p}(n,\F)$, is the maximum number of edges in an $n$-vertex $\F$-free planar graph. Let $Θ_k$ be the family of Theta graphs on $k\geq 4$ vertices, that is, graphs obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Lan, Shi and Song determined an upper bound $\text{ex}_{\mathcal{P}}(n,Θ_6)\leq \frac{18}{7}n-\frac{36}{7}$, but for large $n$, they did not verify that the bound is sharp. In this paper, we improve their bound by proving $\text{ex}_{\mathcal{P}}(n,Θ_6)\leq \frac{18}{7}n-\frac{48}{7}$ and then we demonstrate the existence of infinitely many positive integer $n$ and an $n$-vertex $Θ_6$-free planar graph attaining the bound.

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arXiv2023

The planar Turán number of the seven-cycle

The planar Turán number, $ex_\mathcal{P}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_4)$ and $ex_\mathcal{P}(n,C_5)$. Later on, D. Ghosh et al. obtained sharp upper bound of $ex_\mathcal{P}(n,C_6)$ and proposed a conjecture on $ex_\mathcal{P}(n,C_k)$ for $k\geq 7$. In this paper, we give a sharp upper bound $ex_\mathcal{P}(n,C_7)\leq {18\over 7}n-{48\over 7}$, which satisfies the conjecture of D. Ghosh et al. It turns out that this upper bound is also sharp for $ex_\mathcal{P}(n,\{K_4,C_7\})$, the maximum number of edges in an $n$-vertex planar graph which does not contain $K_4$ or $C_7$ as a subgraph.

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arXiv2023

The planar Turán number of $\{K_4,C_5\}$ and $\{K_4,C_6\}$

Let $\mathcal{H}$ be a set of graphs. The planar Turán number, $ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a subgraph. When $\mathcal{H}=\{H\}$ has only one element, we usually write $ex_\mathcal{P}(n,H)$ instead. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_5)$ and $ex_\mathcal{P}(n,K_4)$. Later on, we obtained sharper bound for $ex_\mathcal{P}(n,\{K_4,C_7\})$. In this paper, we give upper bounds of $ex_\mathcal{P}(n,\{K_4,C_5\})\leq {15\over 7}(n-2)$ and $ex_\mathcal{P}(n,\{K_4,C_6\})\leq {7\over 3}(n-2)$. We also give constructions which show the bounds are tight for infinitely many graphs.

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arXiv2023

The Minimum Number of $4$-Cycles in a Maximal Planar Graph with Small Number of Vertices

Hakimi and Schmeichel determined a sharp lower bound for the number of cycles of length 4 in a maximal planar graph with $n$ vertices, $n\geq 5$. It has been shown that the bound is sharp for $n = 5,12$ and $n\geq 14$ vertices. However, the authors only conjectured the minimum number of cycles of length 4 for maximal planar graphs with the remaining small vertex numbers. In this note, we confirm their conjecture.

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arXiv2023

Extremal problems of double stars

In a generalized Turán problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.

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arXiv2023

The maximum Wiener index of a uniform hypergraph

The Wiener index of a (hyper)graph is calculated by summing up the distances between all pairs of vertices. We determine the maximum possible Wiener index of a connected $n$-vertex $k$-uniform hypergraph and characterize for every~$n$ all hypergraphs attaining the maximum Wiener index.

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arXiv2023

On the rainbow planar Turán number of paths

An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$, known as the rainbow Turán problem, was initiated by Keevash, Mubayi, Sudakov and Verstraëte. We investigate a variation of this problem with the additional restriction that the graph is planar, and we denote the corresponding extremal number by $\ex_{\p}^*(n,F)$. In particular, we determine $\ex_{\p}^*(n,P_5)$, where $P_5$ denotes the $5$-vertex path.

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arXiv2022

The maximum number of cliques in graphs with bounded odd circumference

In this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo's recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

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arXiv2022

The Maximum Number of Pentagons in a Planar Graph

In 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an $n$-vertex planar graph. They precisely determined the maximum number of triangles and $4$-cycles and presented a conjecture for the maximum number of pentagons. In this work, we confirm their conjecture. Even more, we characterize the $n$-vertex, planar graphs with the maximum number of pentagons.

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arXiv2022

Linear three-uniform hypergraphs with no Berge path of given length

Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an $n$-vertex $3$-uniform linear hypergraph, without a Berge path of length $k$ as a subgraph is at most $\frac{(k-1)}{6}n$ for $k\geq 4$.

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arXiv2022

Extremal results for graphs avoiding a rainbow subgraph

We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding a rainbow path of length three.

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arXiv2022

Edges not covered by monochromatic bipartite graphs

Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Turán number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$. Keevash and Sudakov proved that $f(n,H)=ex(n,H)$ if $H$ is an edge-critical graph or $C_4$ and asked if this equality holds for any graph $H$. All known exact values of this question require $H$ to contain at least one cycle. In this paper we focus on acyclic graphs and have the following results: (1) We prove $f(n,H)=ex(n,H)$ when $H$ is a spider or a double broom. (2) A \emph{tail} in $H$ is a path $P_3=v_0v_1v_2$ such that $v_2$ is only adjacent to $v_1$ and $v_1$ is only adjacent to $v_0,v_2$ in $H$. We obtain a tight upper bound for $f(n,H)$ when $H$ is a bipartite graph with a tail. This result provides the first bipartite graphs which answer the question of Keevash and Sudakov in the negative. (3) Liu, Pikhurko and Sharifzadeh asked if $f_k(n,T)=(k-1)ex(n,T)$ when $T$ is a tree. We provide an upper bound for $f_{2k}(n,P_{2k})$ and show it is tight when $2k-1$ is prime. This provides a negative answer to their question.

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arXiv2022

Generalized Turan number for the edge blow-up graph

Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^p$ of $H$ is the graph obtained from replacing each edge in $H$ by a copy of $K_p$ where the new vertices of the cliques are all distinct. Let $C_k$ and $P_k$ denote the cycle and path of length $k$, respectively. In this paper, we find sharp upper bounds for $ex(n,K_3,C_3^3)$ and the exact value for $ ex(n,K_3,P_3^3)$ and determine the graphs attaining these bounds.

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arXiv2022

Generalized outerplanar Turán number of short paths

Let $H$ be a graph. The generalized outerplanar Turán number of $H$, denoted by $f_{\mathcal{OP}}(n,H)$, is the maximum number of copies of $H$ in an $n$-vertex outerplanar graph. Let $P_k$ be the path on $k$ vertices. In this paper we give an exact value of $f_{\mathcal{OP}}(n,P_4)$ and a best asymptotic value of $f_{\mathcal{OP}}(n,P_5)$. Moreover, we characterize all outerplanar graphs containing $f_{\mathcal{OP}}(n,P_4)$ copies of $P_4$.

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arXiv2022

Extremal planar graphs with no cycles of particular lengths

In this paper we estimate the planar Turán number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when $H=C_5$, and study the cases when $G$ is bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly new applications or variants of the "contribution method" introduced by Ghosh, Győri, Martin, Paulos and Xiao in arXiv:2004.14094.

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arXiv2022

The maximum number of triangles in $F_k$-free graphs

The generalized Turán number $ex(n,K_s,H)$ is the maximum number of complete graph $K_s$ in an $H$-free graph on $n$ vertices. Let $F_k$ be the friendship graph consisting of $k$ triangles. Erdős and Sós (1976) determined the value of $ex(n,K_3,F_2)$. Alon and Shikhelman (2016) proved that $ex(n,K_3, F_k)\le (9k-15)(k+1)n.$ In this paper, by using a method developed by Chung and Frankl in hypergraph theory, we determine the exact value of $ex(n,K_3,F_k)$ and the extremal graph for any $F_k$ when $n\ge 4k^3$.

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