arXiv subjects

Morteza Saghafian

Publications and source records attributed to Morteza Saghafian.

25 records · Page 2Linked to original sources

arXiv2022

On the Size of Chromatic Delaunay Mosaics

Given a locally finite set $A \subseteq \mathbb{R}^d$ and a coloring $χ\colon A \to \{0,1,\ldots,s\}$, we introduce the chromatic Delaunay mosaic of $χ$, which is a Delaunay mosaic in $\mathbb{R}^{s+d}$ that represents how points of different colors mingle. Our main results are bounds on the size of the chromatic Delaunay mosaic, in which we assume that $d$ and $s$ are constants. For example, if $A$ is finite with $n = \#{A}$, and the coloring is random, then the chromatic Delaunay mosaic has $O(n^{\lceil{d/2}\rceil})$ cells in expectation. In contrast, for Delone sets and Poisson point processes in $\mathbb{R}^d$, the expected number of cells within a closed ball is only a constant times the number of points in this ball. Furthermore, in $\mathbb{R}^2$ all colorings of a dense set of $n$ points have chromatic Delaunay mosaics of size $O(n)$. This encourages the use of chromatic Delaunay mosaics in applications.

math.CO↗

arXiv2022

On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane

For a locally finite set in $\mathbb{R}^2$, the order-$k$ Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in $k$. As an example, a stationary Poisson point process in $\mathbb{R}^2$ is locally finite, coarsely dense, and generic with probability one. For such a set, the distribution of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-$1$ Delaunay mosaics given by Miles in 1970.

math.CO↗

arXiv2018

On $k$-tuple and $k$-tuple total domination numbers of regular graphs

Let $G$ be a connected graph of order $n$, whose minimum vertex degree is at least $k$. A subset $S$ of vertices in $G$ is a $k$-tuple total dominating set if every vertex of $G$ is adjacent to at least $k$ vertices in $S$. The minimum cardinality of a $k$-tuple total dominating set of $G$ is the $k$-tuple total domination number of $G$, denoted by $γ_{\times k,t}(G)$. Henning and Yeo in \cite{hen} proved that if $G$ is a cubic graph different from the Heawood graph, $γ_{\times 2, t}(G) \leq \frac{5}{6}n$, and this bound is sharp. Similarly, a $k$-tuple dominating set is a subset $S$ of vertices of $G$, $V (G)$ such that $|N[v] \cap S| \geq k$ for every vertex $v$, where $N[v] = \{v\}\cup \{u \in V(G) : uv \in E(G)\}$. The $k$-tuple domination number of $G$, denoted by $γ_{\times k}(G)$, is the minimum cardinality of a $k$-tuple dominating set of $G$. In this paper, we give a simple approach to compute an upper bound for $(r-1)$-tuple total domination number of $r$-regular graphs. Also, we give an upper bound for the $r$-tuple dominating number of $r$-regular graphs. In addition, our method gives algorithms to compute dominating sets with the given bounds, while the previous methods are existential.

math.CO↗

arXiv2017

Chromatic Number and Dichromatic Polynomial of Digraphs

Let $G$ be a graph of order $n$. It is well-known that $α(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $α(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ α(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $α(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $χ(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(Δ^+(D),Δ^-(D))$. We give a short and simple proof for this result. Next, we investigate the chromatic number of tournaments and determine the unique tournament such that for every integer $k>1$, the number of proper $k$-colorings of that tournament is maximum among all strongly connected tournaments with the same number of vertices. Also, we find the chromatic polynomial of the strongly connected tournament with the minimum number of cycles.

math.CO↗

arXiv2017

Is there any polynomial upper bound for the universal labeling of graphs?

A {\it universal labeling} of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ and $u$ in the oriented graph are different from each other. The {\it universal labeling number} of a graph $G$ is the minimum number $k$ such that $G$ has {\it universal labeling} from $\{1,2,\ldots, k\}$ denoted it by $\overrightarrow{χ_{u}}(G) $. We have $2Δ(G)-2 \leq \overrightarrow{χ_{u}} (G)\leq 2^{Δ(G)}$, where $Δ(G)$ denotes the maximum degree of $G$. In this work, we offer a provocative question that is:" Is there any polynomial function $f$ such that for every graph $G$, $\overrightarrow{χ_{u}} (G)\leq f(Δ(G))$?". Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree $T$, $\overrightarrow{χ_{u}}(T)=\mathcal{O}(Δ^3) $. Next, we show that for a given 3-regular graph $G$, the universal labeling number of $G$ is 4 if and only if $G$ belongs to Class 1. Therefore, for a given 3-regular graph $G$, it is an $ \mathbf{NP} $-complete to determine whether the universal labeling number of $G$ is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem.

math.CO↗

arXiv2016

Improving the Bounds On Murty_Simon Conjecture

A graph is said to be diameter-$k$-critical if its diameter is $k$ and removal of any of its edges increases its diameter. A beautiful conjecture by Murty and Simon, says that every diameter-2-critical graph of order $n$ has at most $\lfloor n^2/4\rfloor$ edges and equality holds only for $K_{\lceil n/2 \rceil,\lfloor n/2 \rfloor }$. Haynes et al. proved that the conjecture is true for $Δ\geq 0.7n$. They also proved that for $n>2000$, if $Δ\geq 0.6789n$ then the conjecture is true. We will improve this bound by showing that the conjecture is true for every $n$ if $Δ\geq\ 0.676n$.

math.CO↗

arXiv2012

On a Bounded Budget Network Creation Game

We consider a network creation game in which each player (vertex) has a fixed budget to establish links to other players. In our model, each link has unit price and each agent tries to minimize its cost, which is either its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two versions of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between the vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between the vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. We take the social cost of the created network to be its diameter, and next we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. When the sum of players' budgets is n-1, the equilibrium graphs are always trees, and we prove that their maximum diameter is Theta(n) and Theta(log n) in MAX and SUM versions, respectively. When each vertex has unit budget (i.e. can establish link to just one vertex), the diameter of any equilibrium graph in either version is Theta(1). We give examples of equilibrium graphs in the MAX version, such that all vertices have positive budgets and yet the diameter is Omega(sqrt(log n)). This interesting (and perhaps counter-intuitive) result shows that increasing the budgets may increase the diameter of equilibrium graphs and hence deteriorate the network structure. Then we prove that every equilibrium graph in the SUM version has diameter 2^O(sqrt(log n)). Finally, we show that if the budget of each player is at least k, then every equilibrium graph in the SUM version is k-connected or has diameter smaller than 4.

cs.GT↗