arXiv subjects

Nazli Besharati

Publications and source records attributed to Nazli Besharati.

7 recordsLinked to original sources

arXiv2025

Global coalition sets in graphs

Let $G=(V,E)$ be a graph. A subset $S \subseteq V$ is called a global dominating set of $G$, if it serves as a dominating set in both $G$ and its complement $\overline{G}$. We define two disjoint subsets $V_1,V_2 \subseteq V$ to form a global coalition if neither $V_1$ nor $V_2$ individually constitutes a global dominating set, yet their union $V_1 \cup V_2$ does. A global coalition partition (abbreviated as $gc$-partition) of $G$ is a vertex partition $\pi$ of $V(G)$ such that for every subset $V_i \in \pi$, there exists another subset $V_j \in \pi$ with which $V_i$ forms a global coalition. In this paper, we initiate the study of global coalition in graphs. Specifically, we prove that every graph admits a gc-partition. Additionally, we establish an upper bound on the number of global coalitions in which each member of a gc-partition can participate. We also explore the relationships between global coalition and coalition, as well as between global coalition and perfect coalition in graphs. Finally, we explore properties of $gc$-partitions in unicyclic graphs.

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arXiv2025

Edge Coalitions in Graphs

Coalition concepts have been extensively studied in domination theory for vertex sets, whereas their edge counterparts have remained largely unexplored. Motivated by this, we introduce the notions of edge coalition, edge coalition partition, edge coalition number, and edge coalition graph. We prove that every graph admits an edge coalition partition and establish fundamental properties of these concepts. We derive sharp bounds for the edge coalition number, characterize graphs attaining its extremal values, and determine this parameter for several important graph classes, including complete graphs, complete bipartite graphs, paths, cycles, stars, trees, and unicyclic graphs. We further introduce the edge coalition graph associated with an edge coalition partition and investigate its structural properties. In particular, we characterize the edge coalition graphs of several graph classes and identify all self-edge coalition graphs. These results extend coalition theory from vertices to edges and provide a foundation for further research on edge coalition structures.

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arXiv2023

Edge coalitions in graphs

An edge coalition in a graph $G=(V,E)$ consists of two disjoint sets of edges $E_1$ and $E_2$, neither of which is an edge dominating set but whose union $E_1\cup E_2$ is an edge dominating set. An edge coalition partition in a graph $G$ of order $n=|V|$ and size $m$ is an edge partition $\pi=\{E_1,\cdots,E_k\}$ so that every set $E_i$ of $\pi$ either is a singleton edge dominating set, or is not an edge dominating set but forms an edge coalition with another set $E_j$ which is not an edge dominating set. In this paper we introduce the concept of edge coalition and show that there exists edge coalition for some graphs and trees. The graphs $G$ with small and size number of edge coalition are characterized. Finally, coalition graphs of special graphs are studied.

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arXiv2018

Determination of the size of defining set for Steiner triple systems

Every Steiner triple system is a uniform hypergraph. The coloring of hypergraph and its special case Steiner triple systems, {STS}$(v)$, is studied extensively. But the defining set of the coloring of hypergraph even its special case {STS}$(v)$, is not explored yet. We study minimum defining set and the largest minimal defining set for $3$-coloring of {STS}$(v)$. We determined minimum defining set and the largest minimal defining set, for all non-isomorphic {STS}$(v)$, $v\le 15$. Also we have found the {\sf defining number} for all Steiner triple systems of order $v$, and some lower bounds for the size of the largest minimal defining set for all Steiner triple systems of order $v$, for each admissible $v$.

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arXiv2016

On the chromatic number of Latin square graphs

The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some other classes of Latin squares. With a computer, we find the chromatic number for all main classes of Latin squares of order at most eight.

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arXiv2012

Silver block intersection graphs of Steiner 2-designs

For a block design $\cal{D}$, a series of {\sf block intersection graphs} $G_i$, or $i$-{\rm BIG}($\cal{D}$), $i=0, ..., k$ is defined in which the vertices are the blocks of $\cal{D}$, with two vertices adjacent if and only if the corresponding blocks intersect in exactly $i$ elements. A silver graph $G$ is defined with respect to a maximum independent set of $G$, called a {\sf diagonal} of that graph. Let $G$ be $r$-regular and $c$ be a proper $(r + 1)$-coloring of $G$. A vertex $x$ in $G$ is said to be {\sf rainbow} with respect to $c$ if every color appears in the closed neighborhood $N[x] = N(x) \cup \{x\}$. Given a diagonal $I$ of $G$, a coloring $c$ is said to be silver with respect to $I$ if every $x\in I$ is rainbow with respect to $c$. We say $G$ is {\sf silver} if it admits a silver coloring with respect to some $I$. We investigate conditions for 0-{\rm BIG}($\cal{D}$) and 1-{\rm BIG}($\cal{D}$) of Steiner systems ${\cal{D}}=S(2,k,v)$ to be silver.

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arXiv2011

Independence number of generalized Petersen graphs

Determining the size of a maximum independent set of a graph $G$, denoted by $α(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $α(G)$ for special classes of graphs. This paper is aimed toward studying this problem for the class of generalized Petersen graphs. We find new upper and lower bounds and some exact values for $α(P(n,k))$. With a computer program we have obtained exact values for each $n<78$. In \cite{MR2381433} it is conjectured that $β(P(n, k)) \leq n + \lceil\frac{n}{5}\rceil $, for all $n$ and $k$. We prove this conjecture for some cases. In particular, we show that if $ n> 3k$, the conjecture is valid. We checked the conjecture with our table for $n < 78$ and it had no inconsistency. Finally, we show that for every fix $k$, $α(P(n, k))$ can be computed using an algorithm with running time O(n).

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