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Let G be a graph. It is well-known that G contains a proper vertex-coloring with χ(G) colors with the property that at least one color class of the coloring is a dominating set in G. Among all such proper vertex-coloring of the vertices of G, a coloring with the maximum number of color classes that are dominating sets in G is called a dominating-χ-coloring of G. The number of color classes that are dominating sets in a dominating-χ-coloring of G is defined to be the dominating-χ-color number of G and is denoted by dχ(G). In this paper, we prove that if G is a claw-free graph with minimum degree at least two, then dχ(G)≥2  

Let R be a commutative ring with unity and R+, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ℂAY(R) and GR, the Cayley graph Cay(R+, Z*(R)) and the unitary Cayley graph Cay(R+, U(R)), respectively. For an Artinian ring R, Akhtar et al. (2009) studied GR. In this article, we study ℂAY(R) and determine the clique number, chromatic number, edge chromatic number, domination number, and the girth of ℂAY(R). We also characterize all rings R whose ℂAY(R) is planar. Moreover, we determine all finite rings R whose ℂAY(R) is strongly regular. We prove that ℂAY(R) is strongly regular if and only if it is edge... 

Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals is finite. Among other results, it is shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is... 

Let R be a commutative ring with unity and R+ and Z*(R) be the additive group and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) the Cayley graph Cay(R+, Z*(R)). In this article, we study ℂ𝔸𝕐(R). Among other results, it is shown that for every zero-dimensional nonlocal ring R, ℂ𝔸𝕐(R) is a connected graph of diameter 2. Moreover, for a finite ring R, we obtain the vertex connectivity and the edge connectivity of ℂ𝔸𝕐(R). As a result, ℂ𝔸𝕐(R) gives an algebraic construction for vertex transitive graphs of maximum connectivity. In addition, we characterize all zero-dimensional semilocal... 

In this paper, using the chromatic properties of power graphs we propose a new approach for placing resources in symmetric networks. Our novel placement scheme guarantees a perfect placement when such a solution is feasible in the topology. © Springer-Verlag Berlin Heidelberg 2007  

Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and term rank of G, by rk (G) and Rk (G), respectively. van Nuffelen conjectured that for any graph G, χ (G) ≤ rk (G). The first counterexample to this conjecture was obtained by Alon and Seymour. In 2002, Fishkind and Kotlov proved that for any graph G, χ (G) ≤ Rk (G). Here we improve this upper bound and show that χl (G) ≤ (rk (G) + Rk (G)) / 2, where χl (G) is the list chromatic number of G. © 2006 Elsevier B.V. All rights reserved  

Let R be a ring and X R be a non-empty set. The regular graph of X, Γ(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Γ( GLn(F)) finite? In this paper, we show that if G is a soluble subgroup of GLn(F), then χ(Γ(G))<∞. Also, we show that for every field F, χ(Γ( Mn(F)))=χ(Γ( Mn(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Γ(), where denotes the subgroup generated by A∈ GLn(F)  

In this paper, using the chromatic properties of power graphs we propose a new approach for placing resources in symmetric networks. Our novel placement scheme guarantees a perfect placement when such a solution is feasible in the topology, while in general it answers the question of k-resource placement at a distance d where each non-resource node is able to access k resource nodes within at most d hops away. We define a quasi-perfect graph as a graph whose clique number and chromatic number are equal. We derive important properties of quasi-perfect graphs and use them to find a solution for the resource placement problem. We have also applied the proposed method to find a distant resource... 

Let d (σ) stand for the defining number of the colouring σ. In this paper we consider dmin = minγ d (γ) and dmax = maxγ d (γ) for the onto χ-colourings γ of the circular complete graph Kn, d. In this regard we obtain a lower bound for dmin (Kn, d) and we also prove that this parameter is asymptotically equal to χ - 1. Also, we show that when χ ≥ 4 and s ≠ 0 then dmax (Kχ d - s, d) = χ + 2 s - 3, and, moreover, we prove an inequality relating this parameter to the circular chromatic number for any graph G. © 2006 Elsevier B.V. All rights reserved  

A class G of simple graphs is said to be girth-closed (odd-girth-closed) if for any positive integer g there exists a graph G ∈ G such that the girth (odd-girth) of G is ≥ g. A girth-closed (odd-girth-closed) class G of graphs is said to be pentagonal (odd-pentagonal) if there exists a positive integer g∗ depending on G such that any graph G ∈ G whose girth (odd-girth) is greater than g∗ admits a homomorphism to the five cycle (i.e. is C5-colourable). Although, the question “Is the class of simple 3-regular graphs pentagonal?” proposed by Nešetřil (Taiwan J Math 3:381–423, 1999) is still a central open problem, Gebleh (Theorems and computations in circular colourings of graphs, 2007) has... 

The energy of a graph G, denoted by E (G), is defined as the sum of the absolute values of all eigenvalues of the adjacency matrix of G. It is proved that E (G) ≥ 2 (n - χ (over(G, -))) ≥ 2 (ch (G) - 1) for every graph G of order n, and that E (G) ≥ 2 ch (G) for all graphs G except for those in a few specified families, where over(G, -), χ (G), and ch (G) are the complement, the chromatic number, and the choice number of G, respectively. © 2007 Elsevier Inc. All rights reserved