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Let (Formula presented.) be a graph. A vertex coloring (Formula presented.) of (Formula presented.) is called (Formula presented.)-vertex coloring if (Formula presented.) for every vertex (Formula presented.) of (Formula presented.), where (Formula presented.) is the set of colors of vertices adjacent to (Formula presented.). Let (Formula presented.) be the maximum number of colors used in an (Formula presented.)-vertex coloring of (Formula presented.). We provide some lower and upper bounds for (Formula presented.) in terms of girth, diameter or size of (Formula presented.). Also, for every tree (Formula presented.), we obtain (Formula presented.). © 2018 World Scientific Publishing Company... 

Let G be a graph. A vertex coloring ψ of G is called N2-vertex coloring if |ψ(v)|≤ 2 for every vertex v of G, where ψ(v) is the set of colors of vertices adjacent to v. Let t2(G) be the maximum number of colors used in an N2-vertex coloring of G. We provide some lower and upper bounds for t2(G) in terms of girth, diameter or size of G. Also, for every tree T, we obtain t2(T). © 2018 World Scientific Publishing Company  

In this paper we investigate two conjectures proposed in (Graphs Combin. 13 (1997) 305-314). The first one is uniquely totally colorable (UTC) conjecture which states: Empty graphs, paths, and cycles of order 3k, k a natural number, are the only UTC graphs. We show that if G is a UTC graph of order n, then Δn/2+1, where Δ is the maximum degree of G. Also there is another question about UTC graphs that appeared in (Graphs Combin. 13 (1997) 305-314) as follows: If a graph G is UTC, is it true that in the proper total coloring of G, each color is used for at least one vertex? We prove that if G is a UTC graph of order n and in the proper total coloring of G, there exists a color which did not... 

Scheduling sports leagues has drawn significant attention to itself in recent years, as it involves considerable revenue as well as challenging combinatorial optimization problems. A particular class of these problems is the Traveling Tournament Problem (TTP) which focuses on minimizing the total traveling distance for teams. In this paper, an efficient simulated annealing approach is presented for TTP which applies two simultaneous and disparate models for the problem in order to search the solutions space more effectively. Also, a computationally efficient modified greedy scheme is proposed for constructing a favorable initial solution for the simulated annealing algorithm. Our... 

Caching the popular files in the memories of small base stations (SBSs) is an effective solution to the limited capacity of wireless backhaul links. The efficiency of caching depends on the proper design of the content placement. This is more challenging when the SBSs have overlapped coverage and especially when the users are also mobile. In this letter, we exploit the concept of graph-coloring and model the connection between SBSs and users as well as user mobilities, through a weighted graph in which SBSs are its vertices, and propose some algorithms to color it effectively. Then, we fill the memory of colored SBSs with popular content accordingly. © 1967-2012 IEEE  

In this paper, we combine a novel Sequential Graph Coloring Heuristic Algorithm (SGCHA) with a non-systematic method based on a cultural algorithm to solve the graph coloring problem (GCP). The GCP involves finding the minimum number of colors for coloring the graph vertices such that adjacent vertices have distinct colors. In our solving approach, we first use an estimator which is implemented with SGCHA to predict the minimum colors. Then, in the non-systematic part which has been designed using cultural algorithms, we improve the prediction. Various components of the cultural algorithm have been implemented to solve the GCP with a self adaptive behavior in an efficient manner. As a result... 

Data traffic is increasing in recent years and small cell densification is a promising solution for handling the consequences of this growth. However, it faces with backhual' constraint which limits its efficiency. Caching the popular files near to the small base stations (SBSs) is an effective technique which decreases usage of backhual links. In this paper, we model the network by a graph and propose a new popular files' placement based on graph coloring for which different powerful solutions and algorithms have been presented in the literature. In this respect, we propose an algorithm to fill the caches of SBSs with the popular files and show the efficiency of our proposed algorithm... 

Improving testability during the early stages of High-Level Synthesis (HLS) has several benefits, including reduced test hardware overhead, reduced test costs, reduced design iteration, and significant improved fault coverage. In this paper, we present a novel register allocation method, which is based on weighted graph coloring algorithm, targeting testability improvement for digital circuits. The topics covered in this paper include an overview of HLS and testability parameters, our testability model and experimental results  

Cognitive Radio (CR) is a promised solution to the lack of spectrum, as well as spectrum inefficiency, in current communication networks. A major demand of this technology is for fair spectrum assignment to unlicensed (secondary) users. This paper addresses a new approach to this challenge in CR networks, based on the Graph Coloring Problem (GCP) and the Ant Colony System (ACS). In addition, we compare the performance of the proposed algorithm with Color Sensitive Graph Coloring (CSGC) and Particle Swarm Optimization (PSO) based techniques  

We present a constant-time randomized distributed algorithms in the congested clique model that computes an O(α)-vertex-coloring, with high probability. Here, α denotes the arboricity of the graph, which is, roughly speaking, the edge-density of the densest subgraph. Congested clique is a well-studied model of synchronous message passing for distributed computing with all-to-all communication: per round each node can send one O(log n)-bit message algorithm to each other node. Our O(1)-round algorithm settles the randomized round complexity of the O(α)-coloring problem. We also explain that a similar method can provide a constant-time randomized algorithm for decomposing the graph into O(α)... 

A lucky labeling of a graph G is a function l:V(G)→N, such that for every two adjacent vertices v and u of G, σ w∼vl(w)≠ σ w∼ul(w) (x∼y means that x is joined to y). A lucky number of G, denoted by η(G), is the minimum number k such that G has a lucky labeling l:V(G)→{1,⋯,k}. We prove that for a given planar 3-colorable graph G determining whether η(G)=2 is NP-complete. Also for every k≥2, it is NP-complete to decide whether η(G)=k for a given graph G  

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  

Let G be a graph and for any natural number r, χs ′ (G, r) denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, χs ′ (T, 1) ≤ Δ (T) + 1. Here we generalize this result and show that χs ′ (T, 2) ≤ Δ (T) + 1. Moreover, we show that if for any two vertices u and v with maximum degree d (u, v) ≥ 3, then χs ′ (T, 2) = Δ (T). Also for any tree T with Δ (T) ≥ 3 we prove that χs ′ (T, 3) ≤ 2 Δ... 

Ant Colony Optimization (ACO) is a well-known metaheuristic in which a colony of artificial ants cooperate in exploring good solutions to a combinatorial optimization problem. In this paper, an ACO algorithm is presented for the graph coloring problem. This ACO algorithm conforms to Max-Min Ant System structure and exploits a local search heuristic to improve its performance. Experimental results on DIMACS test instances show improvements over existing ACO algorithms of the graph coloring problem  

Suppose we have a family F of sets. For every S∈F, a set D⊆S is a defining set for (F,S) if S is the only element of F that contains D as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares. In this paper, first, we propose the concept of a defining set of a logical formula, and we prove that the computational complexity of such a problem is Σ2-complete. We also show that the computational complexity of the following problem about the defining set of vertex colorings of graphs is Σ2-complete: INSTANCE: A graph G with a vertex coloring c and an integer k.QUESTION: If...