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Ali Behtoei

Publications and source records attributed to Ali Behtoei.

15 recordsLinked to original sources
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.

math.CO↗
arXiv2014

On the Randic and degree distance indices of the Mycielskian of a graph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph into a new graph which is called the Mycielskian of that graph. In this paper we provide some sharp bounds for the Randic index of the Mycielskian graphs. Also, we determine the degree distance index of the Mycielskian of each graph with diameter two.

math.CO↗
arXiv2013

A Bound for the Locating Chromatic Numbers of Trees

Let $f$ be a proper $k$-coloring of a connected graph $G$ and $Π=(V_1,V_2,\ldots,V_k)$ 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 $c_{{}_Π}(v)=(d(v,V_1),d(v,V_2),\ldots,d(v,V_k)),$ where $d(v,V_i)=\min\{d(v,x): x\in V_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $f$ 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 numbers of trees. We provide a counter example to a theorem of Gary Chartrand et al. [G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, P. Zhang, The locating-chromatic number of a graph, Bull. Inst. Combin. Appl. 36 (2002) 89-101] about the locating chromatic numbers of trees. Also, we offer a new bound for the locating chromatic number of trees. Then, by constructing a special family of trees, we show that this bound is best possible.

math.CO↗
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.

math.CO↗
arXiv2012

The Locating Chromatic Number of the Join of Graphs

Let $f$ be a proper $k$-coloring of a connected graph $G$ and $Π=(V_1,V_2,...,V_k)$ 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 $c_{{}_Π}(v):=(d(v,V_1),d(v,V_2),...,d(v,V_k)),$ where $d(v,V_i)=\min\{d(v,x)|x\in V_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $f$ 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 the join of graphs. We show that when $G_1$ and $G_2$ are two connected graphs with diameter at most two, then $\Cchi_{{}_L}(G_1+G_2)=\Cchi_{{}_L}(G_1)+\Cchi_{{}_L}(G_2)$, where $G_1+G_2$ is the join of $G_1$ and $G_2$. Also, we determine the locating chromatic numbers of the join of paths, cycles and complete multipartite graphs.

math.CO↗
arXiv2011

On the locating chromatic number of Kneser 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 Kneser graphs. First, among some other results we show that $\Cchi_{{}_L}(KG(n,2))=n-1$ for all $n\geq 5$. Then, we prove that $\Cchi_{{}_L}(KG(n,k))\leq n-1$, when $n\geq k^2$. Moreover, we present some bounds for the locating chromatic number of odd graphs.

math.CO↗