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Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*-(R), where W*-(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a*‰Rb and b*‰Ra. Let ω(Γ′(R)) and χ(Γ′(R)) denote the clique number and the chromatic number of Γ′(R), respectively. In this paper, we prove that if R is a finite commutative ring, then Γ′(R) is perfect. Also, we prove that if R is a commutative Artinian non-local ring and ω(Γ′(R)) is finite, then χ(Γ′(R)) = ω(Γ′(R)). For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we... 

Let n>k>d be positive integers. The generalized Kneser graph K(n,k,d) is a graph whose vertices are all the subsets of size k in {1,…,n} and two subsets are adjacent if and only if they have less than d elements in common. For d=1 this is the classical Kneser graph whose chromatic number was calculated by Lovász in Lovász (1978). In this article, we use Hadamard matrices to show that for any integer r≥0, the chromatic number of K(2k+2r,k,d) is at most 8(r+d)2 for k≥4(r+d)2−r. This bound improves the previously known upper bounds drastically. © 2019 Elsevier B.V  

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... 

Let r be a ring with unity. the inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ⊂ J or J ⊂ I. In this paper, we show that In(R) is not connected if and only if R ≅ M 2(D) or D 1 × D 2, for some division rings, D, D 1and D 2. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic... 

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = RZ(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2n, where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices  

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper we show if D is a division ring, then the clique number of I(Mn(D)) (n ≥ 2) is n and for any commutative Artinian ring R the clique number and the chromatic number of I(R) are equal to the number of maximal ideals of R. We prove that for every left Noetherian ring R, the clique number of I(R) is finite. For every finite field F, we also determine an independent set of I(Mn(F)) with maximum size. If F is an infinite field, then we prove that the domination number of I(Mn(F)) is infinite. We... 

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... 

A 2-hued coloring of a graph G is a coloring such that, for every vertex v∈V(G) of degree at least 2, the neighbors of v receive at least two colors. The smallest integer k such that G has a 2-hued coloring with k colors is called the 2-hued chromatic number of G, and is denoted by χ2(G). In this paper, we will show that, if G is a regular graph, then χ2(G)-χ(G)≤2log 2(α(G))+3, and, if G is a graph and δ(G)<2, then χ2(G)-χ(G)≤1+4 Δ2δ-1⌉(1+log 2Δ(G)2Δ(G)-δ(G)(α(G))), and in the general case, if G is a graph, then χ2(G)-χ(G)≤2+min α′(G),α(G)+ω(G)2  

Suppose that R is a commutative ring with identity. Let A(R) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A( R)*=A(R)(0) and two distinct vertices I and J are adjacent if and only if IJ=(0). In Behboodi and Rakeei (2011) [8], it was conjectured that for a reduced ring R with more than two minimal prime ideals, girth(AG(R))=3. Here, we prove that for every (not necessarily reduced) ring R, ω(AG(R))<|Min(R)|, which shows that the conjecture is true. Also in this paper, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. Among... 

Let G be a graph and χl (G) denote the list chromatic number of G. In this paper we prove that for every graph G for which the length of each cycle is divisible by l (l ≥ 3), χl (G) ≤ 3. © 2008 Elsevier B.V. All rights reserved  

The energy of a graph G, denoted by E (G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and rank (G) be the rank of the adjacency matrix of G. In this paper we characterize all graphs with E (G) = rank (G). Among other results we show that apart from a few families of graphs, E (G) ≥ 2 max (χ (G), n - χ (over(G, -))), where n is the number of vertices of G, over(G, -) and χ (G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E (G) in terms of rank (G) are given. © 2008 Elsevier B.V. All rights reserved  

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... 

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  

In this paper, we apply some new algebraic no-homomorphism theorems in conjunction with some new chromatic parameters to estimate the circular chromatic number of graphs. To show the applicability of the general results, as a couple of examples, we generalize a well known inequality for the fractional chromatic number of graphs and we also show that the circular chromatic number of the graph obtained from the Petersen graph by excluding one vertex is equal to 3. Also, we focus on the Johnson-Holroyd-Stahl conjecture about the circular chromatic number of Kneser graphs and we propose an approach to this conjecture. In this regard, we introduce a new related conjecture on Kneser graphs and we... 

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  

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  

In this paper we introduce a chromatic parameter, called the fixing chromatic number, which is related to unique colourability of graphs, in the sense that it measures how one can embed the given graph G in G∪Kt by adding edges between G and Kt to make the whole graph uniquely t-colourable. We study some basic properties of this parameter as well as its relationships to some other well-known chromatic numbers as the acyclic chromatic number. We compute the fixing chromatic number of some graph products by applying a modified version of the exponential graph construction. © 2005 Elsevier B.V. All rights reserved  

Circular chromatic number, χc is a natural generalization of chromatic number. It is known that it is NP-hard to determine whether or not an arbitrary graph G satisfies χ(G)= χc(G). In this paper we prove that this problem is NP-hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k ≥ 2 and n ≥ 3, for a given graph G with χ(G)= n, it is NP-complete to verify if χc(G)≤ n -1/k. © 2004 Wiley Periodicals, Inc