arXiv subjects

Addisu Paulos

Publications and source records attributed to Addisu Paulos.

16 recordsLinked to original sources

arXiv2023

On the Halin Tur\'an number of short cycles

A Halin graph is a graph constructed by embedding a tree with no vertex of degree two in the plane and then adding a cycle to join the tree's leaves. The Halin Tur\'an number of a graph $F$, denoted as $\ex_{\hh}(n,F)$, is the maximum number of edges in an $n$-vertex Halin graph. In this paper, we give the exact value of $\ex_{\mathcal{H}}(n,C_4)$, where $C_4$ is a cycle of length 4. We also pose a conjecture for the Halin Tur\'an number of longer cycles.

math.CO↗

arXiv2023

On the rainbow planar Tur\'an 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 Tur\'an problem, was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete. 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.

math.CO↗

arXiv2021

Planar Tur\'an Number of Double Stars

Given a graph $F$, the planar Tur\'an number of $F$, denoted $\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining sharp upper bound for $\text{ex}_{\mathcal{P}}(n,C_4)$ and $\text{ex}_{\mathcal{P}}(n,C_5)$, where $C_4$ and $C_5$ are cycles of length four and five respectively. In this paper we determined an upper bound for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$, $\text{ex}_{\mathcal{P}}(n,S_{2,3})$, $\text{ex}_{\mathcal{P}}(n,S_{2,4})$, $\text{ex}_{\mathcal{P}}(n,S_{2,5})$, $\text{ex}_{\mathcal{P}}(n,S_{3,3})$ and $\text{ex}_{\mathcal{P}}(n,S_{3,4})$, where $S_{m,n}$ is a double star with $m$ and $n$ leafs. Moreover, the bounds for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$ and $\text{ex}_{\mathcal{P}}(n,S_{2,3})$ are sharp.

math.CO↗

arXiv2021

Generalized outerplanar Tur\'an number of short paths

Let $H$ be a graph. The generalized outerplanar Tur\'an 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 $P_4$.

math.CO↗

arXiv2021

Book free $3$-Uniform Hypergraphs

A $k$-book in a hypergraph consists of $k$ Berge triangles sharing a common edge. In this paper we prove that the number of the hyperedges in a $k$-book-free 3-uniform hypergraph on $n$ vertices is at most $\frac{n^2}{8}(1+o(1))$.

math.CO↗

arXiv2021

The Tur\'an Number of the Triangular Pyramid of $3$-Layers

The Tur\'an number of a graph $H$, denoted by $\text{ex}(n, H)$, is the maximum number of edges in an $n$-vertex graph that does not have $H$ as a subgraph. Let $TP_k$ be the triangular pyramid of $k$-layers. In this paper, we determine that $\text{ex}(n,TP_3)= \frac{1}{4}n^2+n+o(n)$ and pose a conjecture for $\text{ex}(n,TP_4)$.

math.CO↗

arXiv2020

Planar Tur\'an Number of the $\Theta_6$

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 Tur\'an number} of $\F$, denoted by $\ex_{\p}(n,\F)$, is the maximum number of edges in an $n$-vertex $\F$-free planar graph. Let $\Theta_k$ 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 $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{36}{7}$, but for large $n$, they did not verify that the bound is sharp. In this paper, we improve their bound by proving $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{48}{7}$ and then we demonstrate the existence of infinitely many positive integer $n$ and an $n$-vertex $\Theta_6$-free planar graph attaining the bound.

math.CO↗

arXiv2020

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.

math.CO↗

arXiv2020

Planar Tur\'an number of the 6-cycle

Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well studied function, the planar Tur\'an number of $H$, denoted by ${\rm ex}_{\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${\rm ex}_{\mathcal{P}}(n,C_4)$ and ${\rm ex}_{\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${\rm ex}_{\mathcal{P}}(n,C_6) \leq \frac{5}{2}n-7$, for all $n\geq 18$, which improves Lan's result. We also pose a conjecture on ${\rm ex}_{\mathcal{P}}(n,C_k)$, for $k\geq 7$.

math.CO↗

arXiv2020

The Maximum Number of Paths of Length Four in a Planar Graph

Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path on $k$ vertices, is $n^{{\lfloor{\frac{k-1}{2}}\rfloor}+1}$. In this paper we determine the asymptotic value of $f(n,P_5)$ and give conjectures for longer paths.

math.CO↗

arXiv2020

The maximum number of induced $C_5$'s in a planar graph

Finding the maximum number of induced cycles of length $k$ in a graph on $n$ vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidick\'{y} and Pfender answered the question in the case $k=5$. In this paper we determine precisely, for all sufficiently large $n$, the maximum number of induced $5$-cycles that an $n$-vertex planar graph can contain.

math.CO↗

arXiv2020

Generalized Planar Tur\'an Numbers

In a generalized Tur\'an problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Tur\'an problems where the host graph is planar. In particular we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geq 1$. We obtain the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.

math.CO↗

arXiv2020

Wiener Index of Quadrangulation Graphs

The Wiener index of a graph $G$, denoted $W(G)$, is the sum of the distances between all pairs of vertices in $G$. \'E. Czabarka, et al. conjectured that for an $n$-vertex, $n\geq 4$, simple quadrangulation graph $G$, \begin{equation*}W(G)\leq \begin{cases} \frac{1}{12}n^3+\frac{7}{6}n-2, &\text{ $n\equiv 0~(mod \ 2)$,}\\ \frac{1}{12}n^3+\frac{11}{12}n-1, &\text{ $n\equiv 1~(mod \ 2)$}. \end{cases} \end{equation*} In this paper, we confirm this conjecture.

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arXiv2019

The Maximum Wiener Index of Maximal Planar Graphs

The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an $n$-vertex maximal planar graph is at most $\lfloor\frac{1}{18}(n^3+3n^2)\rfloor$. We prove this conjecture and for every $n$, $n \geq 10$, determine the unique $n$-vertex maximal planar graph for which this maximum is attained.

math.CO↗

arXiv2019

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.

math.CO↗

arXiv2019

The Maximum Number of Paths of Length Three in a Planar Graph

Let $f(n,H)$ denote the maximum number of copies of $H$ possible in an $n$-vertex planar graph. The function $f(n,H)$ has been determined when $H$ is a cycle of length $3$ or $4$ by Hakimi and Schmeichel and when $H$ is a complete bipartite graph with smaller part of size 1 or 2 by Alon and Caro. We determine $f(n,H)$ exactly in the case when $H$ is a path of length 3.

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