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Circular colouring and algebraic no-homomorphism theorems
, Article European Journal of Combinatorics ; Volume 28, Issue 6, August 2007, Pages 1843–1853 ; Hajiabolhassan, Hossein ; Sharif University of Technology
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...
On defining numbers of circular complete graphs
, Article Discrete Mathematics ; Volume 307, Issue 2, 28 January 2007, Pages 173–180 ; Sharif University of Technology
Abstract
Let d(σ)d(σ) stand for the defining number of the colouring σσ. In this paper we consider View the MathML sourcedmin=minγd(γ) and View the MathML sourcedmax=maxγd(γ) for the onto χχ-colourings γγ of the circular complete graph Kn,dKn,d. In this regard we obtain a lower bound for dmin(Kn,d)dmin(Kn,d) and we also prove that this parameter is asymptotically equal to χ-1χ-1. Also, we show that when χ⩾4χ⩾4 and s≠0s≠0 then dmax(Kχd-s,d)=χ+2s-3dmax(Kχd-s,d)=χ+2s-3, and, moreover, we prove an inequality relating this parameter to the circular chromatic number for any graph G
On connected colourings of graphs
, Article Ars Combinatoria ; Volume 89 , 2008 , Pages 115-126 ; 03817032 (ISSN) ; Hajiabolhassan, H ; Hamedazimi, N ; Sharif University of Technology
2008
Abstract
In this paper, first we introduce the concept of a connected graph homomorphism as a homomorphism for which the inverse image of any edge is either empty or a connected graph, and then we concentrate on chromatically connected (resp. chromatically disconnected) graphs such as G for which any χ(G)-colouring is a connected (resp. disconnected) homomorphism to K χ(G). In this regard, we consider the relationships of the new concept to some other notions as uniquely-colourability. Also, we specify some classes of chromatically disconnected graphs such as Kneser graphs KG(m, n) for which m is sufficiently larger than n, and the line graphs of non-complete class II graphs. Moreover, we prove that...
On small uniquely vertex-colourable graphs and Xu's conjecture [electronic resource]
, Article Discrete Mathematics ; Volume 223, Issues 1–3, 28 August 2000, Pages 93–108 ; Naserasr, Reza ; Sharif University of Technology
Abstract
Consider the parameter Λ(G) = |E(G)| - |V(G)|(k - 1) + (k2) for a k-chromatic graph G, on the set of vertices V(G) and with the set of edges E(G). It is known that Λ(G)≥0 for any k-chromatic uniquely vertex-colourable graph G (k-UCG), and, S.J. Xu has conjectured that for any k-UCG, G, Λ(G) = 0 implies that cl(G) = k; in which cl(G) is the clique number of G. In this paper, first, we introduce the concept of the core of a k-UCG as an induced subgraph without any colour-class of size one, and without any vertex of degree k - 1. Considering (k, n)-cores as k-UCGs on n vertices, we show that edge-minimal (k, 2k)-cores do not exist when k ≥ 3, which shows that for any edge-minimal k-UCG on 2k...
Graph homomorphisms through random walks [electronic resource]
, Article Journal of Graph Theory ; 2003, Volume 44, Issue 1, pages 15–38 ; Hajiabolhassan, Hossein ; Sharif University of Technology
Abstract
In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff–Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which also provides a generalization of a theorem of Mohar (1992) as a necessary condition for Hamiltonicity. In particular, in the case that the range is a Cayley graph or an edge-transitive graph, we obtain theorems with a corollary about the existence of homomorphisms to cycles. This, specially, provides...
A hybrid SA/TS algorithm for graph coloring problem [electronic resource]
, Article International Journal of Operational Research (IJOR) ; Volume 11, Number 2/2011 Pages 136-159 ; Eshghi, Kourosh ; Sharif University of Technology
Abstract
The graph colouring problem, as an important NP-complete problem, is considered in this paper and a hybrid meta-heuristic approach is developed to solve it. The initial solution of the algorithm, generated by a heuristic method, is used by a simulated annealing (SA) approach to generate new solutions until no progress in a number of solutions reported. At this stage, the algorithm will use a tabu search routine and this local search operator will be followed for some iterations. After finding a better solution, the algorithm is again followed through SA. Efficiency of the algorithm is showed through various experiments on well-known benchmark problems of DIMACS. Comparison with the available...
An efficient hybrid meta-heuristic ant system for minimum sum colouring problem
, Article International Journal of Operational Research ; Volume 34, Issue 2 , 2019 , Pages 269-284 ; 17457645 (ISSN) ; Eshghi, K ; Sharif University of Technology
Inderscience Enterprises Ltd
2019
Abstract
Graph sum colouring problem is a special class of graph vertex colouring problem. Because of its various applications in practical areas, especially in scheduling, many researchers have been focused on it during the past decade. In recent years, several heuristic and meta-heuristic algorithms have been developed to solve sum colouring problem. In this research, a hybrid algorithm based on mini-max ant system and simulated annealing is applied for the problem. This algorithm is tested on benchmark random graphs and compared to the previous algorithms. Results show that in many cases, the best known results can be obtained or improved by the proposed algorithm. Copyright © 2019 Inderscience...
An efficient hybrid meta-heuristic ant system for minimum sum colouring problem
, Article International Journal of Operational Research ; Volume 34, Issue 2 , 2019 , Pages 269-284 ; 17457645 (ISSN) ; Eshghi, K ; Sharif University of Technology
Inderscience Enterprises Ltd
2019
Abstract
Graph sum colouring problem is a special class of graph vertex colouring problem. Because of its various applications in practical areas, especially in scheduling, many researchers have been focused on it during the past decade. In recent years, several heuristic and meta-heuristic algorithms have been developed to solve sum colouring problem. In this research, a hybrid algorithm based on mini-max ant system and simulated annealing is applied for the problem. This algorithm is tested on benchmark random graphs and compared to the previous algorithms. Results show that in many cases, the best known results can be obtained or improved by the proposed algorithm. Copyright © 2019 Inderscience...
Graph homomorphisms through random walks
, Article Journal of Graph Theory ; Volume 44, Issue 1 , 2003 , Pages 15-38 ; 03649024 (ISSN) ; Hajiabolhassan, H ; Sharif University of Technology
Wiley-Liss Inc
2003
Abstract
In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff-Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which also provides a generalization of a theorem of Mohar (1992) as a necessary condition for Hamiltonicity. In particular, in the case that the range is a Cayley graph or an edge-transitive graph, we obtain theorems with a corollary about the existence of homomorphisms to cycles. This, specially, provides...
On small uniquely vertex-colourable graphs and Xu's conjecture
, Article Discrete Mathematics ; Volume 223, Issue 1-3 , 2000 , Pages 93-108 ; 0012365X (ISSN) ; Naserasr, R ; Sharif University of Technology
Elsevier
2000
Abstract
Consider the parameter Λ(G) = |E(G)| - |V(G)|(k - 1) + (k2) for a k-chromatic graph G, on the set of vertices V(G) and with the set of edges E(G). It is known that Λ(G)≥0 for any k-chromatic uniquely vertex-colourable graph G (k-UCG), and, S.J. Xu has conjectured that for any k-UCG, G, Λ(G) = 0 implies that cl(G) = k; in which cl(G) is the clique number of G. In this paper, first, we introduce the concept of the core of a k-UCG as an induced subgraph without any colour-class of size one, and without any vertex of degree k - 1. Considering (k, n)-cores as k-UCGs on n vertices, we show that edge-minimal (k, 2k)-cores do not exist when k ≥ 3, which shows that for any edge-minimal k-UCG on 2k...
A hybrid algorithm of simulated annealing and tabu search for graph colouring problem
, Article International Journal of Operational Research ; Volume 11, Issue 2 , 2011 , Pages 136-159 ; 17457645 (ISSN) ; Eshghi, K ; Sharif University of Technology
2011
Abstract
The graph colouring problem, as an important NP-complete problem, is considered in this paper and a hybrid meta-heuristic approach is developed to solve it. The initial solution of the algorithm, generated by a heuristic method, is used by a simulated annealing (SA) approach to generate new solutions until no progress in a number of solutions reported. At this stage, the algorithm will use a tabu search routine and this local search operator will be followed for some iterations. After finding a better solution, the algorithm is again followed through SA. Efficiency of the algorithm is showed through various experiments on well-known benchmark problems of DIMACS. Comparison with the available...
Function simulation, graph grammars and colourings
, Article International Journal of Computer Mathematics ; 2013 ; 00207160 (ISSN) ; Rahimi, A ; Taati, S ; Sharif University of Technology
Abstract
We prove that to any partial function φ{symbol} defined on a finite set, there corresponds an infinite class of graphs that could be generated by a graph grammar such that each graph in the class represents the function in the sense that evaluation of the function at any point x of its domain can be simulated by finding the unique extension of a partial vertex colouring of the graph specified by x. We show that in the proposed setup, generating such simulator graphs as well as finding the colouring extensions can be computed effectively in polynomial time. We also discuss some applications of this scenario in producing instances of the graph colouring problem near its phase transition that...
Silver cubes
, Article Graphs and Combinatorics ; Volume 24, Issue 5 , 2008 , Pages 429-442 ; 09110119 (ISSN) ; Goddyn, L.A ; Mahmoodian, E. S ; Verdian Rizi, M ; Sharif University of Technology
2008
Abstract
An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n - 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K n d , I, c) where I is a maximum independent set in a Cartesian power of the complete graph K n , and c : V(K n d)→ {1,2,...,d(n-1)+1} is a vertex colouring where, for ν ∈ I, the closed neighbourhood N[ν] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a...
A novel low-complexity dynamic frequency selection algorithm for cognitive radios
, Article 4th IEEE International Symposium on Wireless Communication Systems 2007, ISWCS'07, Trondheim, 16 October 2007 through 19 October 2007 ; 2007 , Pages 558-562 ; 1424409799 (ISBN); 9781424409792 (ISBN) ; Khalaj, B. H ; Sharif University of Technology
2007
Abstract
In this paper, a non-cooperative distributed frequency selection algorithm is proposed for cognitive radio networks in which in addition to interference minimization, the minimization of implementation complexity and probability of interruption of secondary users are also addressed. The convergence and interference reduction property of the algorithm is also demonstrated using game theoretic concepts. In order to evaluate the performance of the algorithm, the associated optimization problem is formulated into a variant of graph colouring problem, and a centralized algorithm is also proposed as an approximation of the optimal solution. Finally, simulation results demonstrate that the proposed...