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Behnaz Omoomi

Publications and source records attributed to Behnaz Omoomi.

At least 19 recordsLinked to original sources
arXiv2019

Star edge coloring of Cactus graphs

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that no path or cycle of length four is bi-colored. The star chromatic index of $G$, denoted by $χ^{\prime}_{s}(G)$, is the minimum $k$ such that $G$ admits a star edge coloring with $k$ colors. Bezegov{á} et al. (Star edge coloring of some classes of graphs, J. Graph Theory, 81(1), pp.73-82. 2016) conjectured that the star chromatic index of outerplanar graphs with maximum degree $Δ$, is at most $\left\lfloor\frac{3Δ}{2}\right\rfloor+1$. In this paper, we prove this conjecture for a class of outerplanar graphs, namely Cactus graphs, wherein every edge belongs to at most one cycle.

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arXiv2019

Fostering Peer Learning through a New Game-Theoretical Approach in a Blended Learning Environment

Obtaining knowledge and skill achievement through peer learning can lead to higher academic achievement. However, peer learning implementation is not just about putting students together and hoping for the best. At its worst-designed, peer learning may result in one person doing all the effort for instance, or may fail to encourage the students to interact enough with the task and so enhance the task in hand. This study proposes a mechanism as well as an instructional design to foster well-organized peer learning based on game theory $(PD\_PL)$. The proposed mechanism uses prisoner's dilemma and maps the strategy and payoff concepts found in prisoner's dilemma onto a peer learning atmosphere. PD\_PL was implemented during several sessions of four university courses and with 142 computer engineering students. %The results of the pre-test and post-test exams of all the sessions were compared with R software through Paired Hotelling's T-Square analysis in order to investigate the impacts of $PD\_PL$ and the proposed instructional design on students' personal learning. The study results indicated that PD\_PL was beneficial and favourable to the students. Further analysis showed that the $PD\_PL$ had sometimes even enhanced learning by up to $47.2\%$. %The results of a subjective evaluation showed that the majority of respondents found $PD\_PL$ to be an attractive and efficient tool for learning enhancement. %Everybody who is interested in designing peer learning programs and tools will find this study interesting.

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arXiv2019

Clique-coloring of $K_{3,3}$-minor free graphs

A clique-coloring of a given graph $G$ is a coloring of the vertices of $G$ such that no maximal clique of size at least two is monocolored. The clique-chromatic number of $G$ is the least number of colors for which $G$ admits a clique-coloring. It has been proved that every planar graph is $3$-clique colorable and every claw-free planar graph, different from an odd cycle, is $2$-clique colorable. In this paper, we generalize these results to $K_{3,3}$-minor free ($K_{3,3}$-subdivision free) graphs.

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arXiv2018

A Polynomial Time Algorithm to Find the Star Chromatic Index of Trees

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we present a polynomial time algorithm that finds an optimum star edge coloring for every tree. We also provide some tight bounds on the star chromatic index of trees with diameter at most four, and using these bounds we find a formula for the star chromatic index of certain families of trees.

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arXiv2018

Star Edge Coloring of the Cartesian Product of Graphs

A star edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every path and cycle of length four in $G$ uses at least three different colors. The star chromatic index of a graph $G$, is the smallest integer $k$ for which $G$ admits a star edge coloring with $k$ colors. In this paper, we first obtain some upper bounds for the star chromatic index of the Cartesian product of two graphs. We then determine the exact value of the star chromatic index of $2$-dimensional grids. We also obtain some upper bounds on the star chromatic index of the Cartesian product of a path with a cycle, $d$-dimensional grids, $d$-dimensional hypercubes and $d$-dimensional toroidal grids, for every positive integer $d$.

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arXiv2017

Injective chromatic number of outerplanar graphs

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by $χ_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $Δ$ and girth $g$ is studied. It is shown that for every outerplanar graph, $χ_i(G)\leq Δ+2$, and this bound is tight. Then, it is proved that for outerplanar graphs with $Δ=3$, $χ_i(G)\leq Δ+1$ and the bound is tight for outerplanar graphs of girth three and $4$. Finally, it is proved that, the injective chromatic number of $2-$connected outerplanar graphs with $Δ=3$, $g\geq 6$ and $Δ\geq 4$, $g\geq 4$ is equal to $Δ$.

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arXiv2015

A characterization of some graphs with metric dimension two

A set W \subseteq V (G) is called a resolving set, if for each pair of distinct vertices u,v \in V (G) there exists t \in W such that d(u,t) \neq d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dim_M(G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k \leq j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) = deg(v) = j. In this paper, we prove that if G is a k-path, then dim_M(G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two.

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arXiv2015

Sigma clique covering of graphs

The sigma clique cover number (resp. sigma clique partition number) of graph G, denoted by scc(G) (resp. scp(G)), is defined as the smallest integer k for which there exists a collection of cliques of G, covering (resp. partitioning) all edges of G such that the sum of sizes of the cliques is at most k. In this paper, among some results we provide some tight bounds for scc and scp.

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arXiv2014

Pairwise Balanced Designs and Sigma Clique Partitions

In this paper, we are interested in minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers n;m, where m ? n, let S(n;m) be the smallest integer s for which there exists a PBD on n points whose largest block has size m and the sum of its block sizes is equal to s. Also, let S0(n;m) be the smallest integers for which there exists a PBD on n points which has a block of size m and the sum of it block sizes is equal to s. We prove some lower bounds for S(n;m) and S0(n;m). Moreover, we apply these bounds to determine the asymptotic behaviour of the sigma clique partition number of the graph Kn-Km, Cocktail party graphs and complement of paths and cycles.

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arXiv2013

Some lower bounds for the $L$-intersection number of graphs

For a set of non-negative integers $L$, the $L$-intersection number of a graph is the smallest number $l$ for which there is an assignment on the vertices to subsets $A_v \subseteq \{1,\dots, l\}$, such that every two vertices $u,v$ are adjacent if and only if $|A_u \cap A_v|\in L$. The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the vertices in different parts. In this paper, some lower bounds for the (bipartite) $L$-intersection number of a graph for various types $L$ in terms of the minimum rank of graph are obtained.

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arXiv2013

Small oriented cycle double cover of graphs

A small oriented cycle double cover (SOCDC)} of a bridgeless graph $G$ on $n$ vertices is a collection of at most $n-1$ directed cycles of the symmetric orientation, $G_s$, of $G$ such that each edge of $G_s$ lies in exactly one of the cycles. It is conjectured that every 2-connected graph except two complete graphs $K_4$ and $K_6$ has an $\rm SOCDC$. In this paper, we study graphs with $\rm SOCDC$ and obtain some properties of the minimal counterexample to this conjecture.

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arXiv2012

On the Locating Chromatic Number of the Cartesian Product of Graphs

Let $c$ be a proper $k$-coloring of a connected graph $G$ and $Π=(C_1,C_2,...,C_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $Π$ is defined to be the ordered $k$-tuple $c_{{}_Π}(v):=(d(v,C_1),d(v,C_2),...,d(v,C_k)),$ where $d(v,C_i)=\min\{d(v,x) | x\in C_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $c$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $\Cchi_{{}_L}(G)$. In this paper, we study the locating chromatic number of grids, the cartesian product of paths and complete graphs, and the cartesian product of two complete graphs.

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arXiv2012

A generalization of line graphs via link scheduling in wireless networks

In single channel wireless networks, concurrent transmission at different links may interfere with each other. To improve system throughput, a scheduling algorithm is necessary to choose a subset of links at each time slot for data trasmission. Throughput optimal link scheduling discipline in such a wireless network is generally an NP-hard problem. In this paper, we develop a poylnomial time algorithm for link scheduling problem provided that network conflict graph is line multigraph. (i.e. line graph for which its root graph is multigraph). This result can be a guideline for network designers to plan the topology of a stationary wireless network such that the required conditions hold and then the throughput optimal algorithm can be run in a much less time.

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arXiv2012

Local Clique Covering of Graphs

A k-clique covering of a simple graph G, is an edge covering of G by its cliques such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called local clique cover number of G and is denoted by $lcc(G)$. Local clique cover number can be viewed as the local counterpart of the clique cover number which is equal to the minimum total number of cliques covering all edges. In this paper, several aspects of the problem are studied and its relationships to other well-known problems are discussed. Moreover, the local clique cover number of claw-free graphs and its subclasses are notably investigated. In particular, it is proved that local clique cover number of every claw-free graph is at most $cΔ/ \logΔ$, where $Δ$ is the maximum degree of the graph and $c$ is a universal constant. It is also shown that the bound is tight, up to a constant factor. Furthermore, it is established that local clique number of the linear interval graphs is bounded by $\logΔ+ 1/2 \log \logΔ+ O(1)$. Finally, as a by-product, a new Bollobas-type inequality is obtained for the intersecting pairs of set systems.

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arXiv2012

On the oriented perfect path double cover conjecture

An {\sf oriented perfect path double cover} ($\rm OPPDC$) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each edge of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{á} and Ne{š}et{ř}il (Discrete Math. 276 (2004) 287-294) conjectured that every graph except two complete graphs $K_3$ and $K_5$ has an $\rm OPPDC$ and they proved that the minimum degree of the minimal counterexample to this conjecture is at least four. In this paper, among some other results, we prove that the minimal counterexample to this conjecture is 2-connected and 3-edge-connected.

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arXiv2012

On the 1-2-3-conjecture

A k-edge-weighting of a graph G is a function w: E(G)->{1,2,...,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v in V(G), c(v) is sum of weights of the edges that are adjacent to vertex v. If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge weighting (VCk-EW). Karonski et al. (J. Combin. Theory Ser. B 91 (2004) 151-157) conjectured that every graph admits a VC3-EW. This conjecture is known as 1-2-3-conjecture. In this paper, frst, we study the vertex-coloring edge-weighting of the cartesian product of graphs. Among some results, we prove that the 1-2-3-conjecture holds for some infinite classes of graphs. Moreover, we explore some properties of a graph to admit a VC2-EW

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arXiv2012

Uniquely dimensional graphs

A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$ with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we study some properties of uniquely dimensional graphs.

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