Sharif Digital Repository

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)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  

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... 

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 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 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... 

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... 

The square of a graph G denoted by G2, is the graph with the same vertex set as G and edges linking pairs of vertices at distance at most 2 in G. The chromatic number of the square of the Cartesian product of two cycles was previously determined for some cases. In this paper, we determine the precise value of χ((Cm□Cn)2) for all the remaining cases. We show that for all ordered pairs (m,n) except for (7,11) we have χ(Cm□Cn)2)=γV((Cm□Cn)2)|α((Cm□Cn)2), where α(G) denotes the independent number of G. This settles a conjecture of Sopena and Wu (2010). We also show that the smallest integer k such that χ(Cm□Cn2)≤6 for every m,≥k is 10. This answers a question of Shao and Vesel (2013)  

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... 

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... 

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... 

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... 

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  

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 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... 

Electrochromism (EC) is a unique property of certain materials that undergo an electrochemical-induced change in colouration. During the last decades, electrochromic materials (ECMs) have been applied in a variety of technologies ranging from smart windows to information displays and energy storage devices. More recently, ECMs have attracted the attention of the (bio)sensing community thanks to their ability to combine the sensitivity of electrochemical methods with the intuitive readout of optical sensors. Although still a nascent technology, EC-based sensors are on the rise with several targets (e.g. cancer biomarkers, bacteria, metabolites and pesticides), which have already been detected... 

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... 

In this paper, we show that every (3k−3)-edge-connected graph G, under a certain degree condition, can be edge-decomposed into k factors G1,…,Gk such that for each vertex v∈V(Gi), |dGi(v)−dG(v)/k|<1, where 1≤i≤k. As an application, we deduce that every 6-edge-connected graph G can be edge-decomposed into three factors G1, G2, and G3 such that for each vertex v∈V(Gi) with 1≤i≤3, |dGi(v)−dG(v)/3|<1, unless G has exactly one vertex z with dG(z)⁄≡30. Next, we show that every odd-(3k−2)-edge-connected graph G can be edge-decomposed into k factors G1,…,Gk such that for each vertex v∈V(Gi), dGi(v) and dG(v) have the same parity and |dGi(v)−dG(v)/k|<2, where k is an odd positive integer and 1≤i≤k.... 

We revisit the classical edge coloring problem for general graphs in the streaming model. In this model, the input graph is presented as a stream of edges, and the algorithm must report colors assigned to the edges in a streaming fashion, using a memory of size O(n polylog n) on graphs of up to O(n2) edges. In ESA 2019 and SOSA 2021, two elegant randomized algorithms were presented for this problem in the adversarial edge arrival model, where the latest one colors any input graph using O(Δ2/s) colors with high probability in O (ns) space. In this short note, we propose two extremely simple streaming algorithms that achieve the same color and space bounds deterministically. Besides being...