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chromatic-numbers
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On the complexity of the circular chromatic number
, Article Journal of Graph Theory ; Volume 47, Issue 3 , 2004 , Pages 226-230 ; 03649024 (ISSN) ; Tusserkani, R ; Sharif University of Technology
Wiley-Liss Inc
2004
Abstract
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
The regular graph of a non-commutative ring
, Article Electronic Notes in Discrete Mathematics ; Vol. 45, issue , January , 2014 , pp. 79-85 ; ISSN: 15710653 ; Heydari, F ; Sharif University of Technology
Abstract
Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x, y∈R are adjacent if and only if x+y∈Z(R). Let the regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of total graph and regular graph of a commutative ring are contained in the set {3, 4, ∞}. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). Also, we prove that if R is a reduced left Noetherian ring and 2∈Z(R), then the chromatic number and the clique number of Reg(Γ(R)) are the...
The regular graph of a noncommutative ring
, Article Bulletin of the Australian Mathematical Society ; Vol. 89, issue. 1 , February , 2014 , pp. 132-140 ; ISSN: 00049727 ; Heydari, F ; Sharif University of Technology
Abstract
Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by (TΓ (R)) is a graph with all elements of R as vertices, and two distinct vertices x, y in R are adjacent if and only if x + y Z(R). Let the regular graph of R, Reg (Γ(R)), be the induced subgraph of T(Γ (R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set { 3, 4,} . In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if R is a reduced left Noetherian ring and 2 Z(R), then the chromatic number and the clique number of Reg...
Intersection graph of submodules of a module
, Article Journal of Algebra and its Applications ; Volume 11, Issue 1 , 2012 ; 02194988 (ISSN) ; Tavallaee, H. A ; Ghezelahmad, S. K ; Sharif University of Technology
Abstract
Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-module M, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among...
A Class of Weakly Perfect Graphs
, Article Czechoslovak Mathematical Journal ; Volume 60, Issue 4 , 2010 , Pages 1037-1041 ; 00114642 (ISSN) ; Pournaki, M. R ; Yassemi, S ; Sharif University of Technology
Abstract
A graph is called weakly perfect if its chromatic number equals its clique number. In this note a new class of weakly perfect graphs is presented and an explicit formula for the chromatic number of such graphs is given
The coloring of the cozero-divisor graph of a commutative ring
, Article Discrete Mathematics, Algorithms and Applications ; Volume 12, Issue 3 , 2020 ; Khojasteh, S ; Sharif University of Technology
World Scientific
2020
Abstract
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...
On the chromatic number of generalized Kneser graphs and Hadamard matrices
, Article Discrete Mathematics ; Volume 343, Issue 2 , February , 2020 ; Moghaddamzadeh, M. J ; Sharif University of Technology
Elsevier B.V
2020
Abstract
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
Circular colouring and algebraic no-homomorphism theorems
, Article European Journal of Combinatorics ; Volume 28, Issue 6 , 2007 , Pages 1843-1853 ; 01956698 (ISSN) ; Hajiabolhassan, H ; Sharif University of Technology
2007
Abstract
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...
List coloring of graphs having cycles of length divisible by a given number
, Article Discrete Mathematics ; Volume 309, Issue 3 , 2009 , Pages 613-614 ; 0012365X (ISSN) ; Ghanbari, M ; Jahanbekam, S ; Jamaali, M ; Sharif University of Technology
2009
Abstract
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
[r; s; t]-Coloring of Graphs
, M.Sc. Thesis Sharif University of Technology ; Mahmoodian, Ebadollah (Supervisor)
Abstract
Graph coloring is an important concept in graph theory. There have been much developments in this concept recently, vertex coloring, edge coloring and total coloring are studied. But one of the newest kind of graph coloring is [r; s; t] -coloring. This coloring is introduced in 2003 by A. Kemnitz and M. Marangio. In this coloring the difference of any two adjecent vertices, any two adjecent edges, and any adjecent vertex and edge must be at least r, s and t respectively. In this thesis we study [r; s; t] -coloring of graphs and discuss about its results
The regular graph of a commutative ring
, Article Periodica Mathematica Hungarica ; Volume 67, Issue 2 , 2013 , Pages 211-220 ; 00315303 (ISSN) ; Heydari, F ; Sharif University of Technology
2013
Abstract
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
On the idempotent graph of a ring
, Article Journal of Algebra and its Applications ; Volume 12, Issue 6 , September , 2013 ; 02194988 (ISSN) ; Habibi, M ; Majidinya, A ; Manaviyat, R ; Sharif University of Technology
2013
Abstract
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...
On the coloring of the annihilating-ideal graph of a commutative ring
, Article Discrete Mathematics ; Volume 312, Issue 17 , 2012 , Pages 2620-2626 ; 0012365X (ISSN) ; Akbari, S ; Nikandish, R ; Nikmehr, M. J ; Shaveisi, F ; Sharif University of Technology
Elsevier
2012
Abstract
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...
The inclusion ideal graph of rings
, Article Communications in Algebra ; Volume 43, Issue 6 , 2015 , Pages 2457-2465 ; 00927872 (ISSN) ; Habibi, M ; Majidinya, A ; Manaviyat, R ; Sharif University of Technology
Taylor and Francis Inc
2015
Abstract
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 1 and 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...
On the structure of the power graph and the enhanced power graph of a group
, Article Electronic Journal of Combinatorics ; Volume 24, Issue 3 , 2017 ; 10778926 (ISSN) ; Akbari, S ; Cameron, P. J ; Nikandish, R ; Shaveisi, F ; Sharif University of Technology
Abstract
Let G be a group. The power graph of G is a graph with the vertex set G, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence number. For a bounded exponent group, we prove its power graph is a perfect graph and we determine its clique/chromatic number. Furthermore, it is proved that for every group G, the clique number of the power graph of G is at most countably infinite. We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between. We call this new graph the enhanced power graph. For an arbitrary pair of these three graphs we...
Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number
, Article Discrete Applied Mathematics ; Volume 160, Issue 15 , 2012 , Pages 2142-2146 ; 0166218X (ISSN) ; Ahadi, A ; Sharif University of Technology
Elsevier
2012
Abstract
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
Some relations between rank, chromatic number and energy of graphs
, Article Discrete Mathematics ; Volume 309, Issue 3 , 2009 , Pages 601-605 ; 0012365X (ISSN) ; Ghorbani, E ; Zare, S ; Sharif University of Technology
2009
Abstract
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
Unique list-colourability and the fixing chromatic number of graphs
, Article Discrete Applied Mathematics ; Volume 152, Issue 1-3 , 2005 , Pages 123-138 ; 0166218X (ISSN) ; Hajiabolhassan, H ; Sharif University of Technology
2005
Abstract
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
Intersection Graph
, M.Sc. Thesis Sharif University of Technology ; Akbari, Saeed (Supervisor)
Abstract
Let R be a ring with unity, M be a unitary left R-module and I(M)* be the set of all non-trivial submodules of M. The intersection graph of submodules of M, denoted by G(M), is a graph with the vertex set I(M)* and two distinct vertices N and K are adjacent if and only if N\K ̸= 0. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star...
Unique list-colourability and the fixing chromatic number of graphs [electronic resource]
, Article Discrete Applied Mathematics ; Volume 152, Issues 1–3, 1 November 2005, Pages 123–138 ; Hajiabolhassan, Hossein ; Sharif University of Technology
Abstract
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