Minimum sum edge colorings of multicycles

crab.rutgers.edu

2011

In this paper we are interested in the elaboration of an approached solution to the sum coloring problem (MSCP), which is an NP-hard problem derived from the graphs coloring (GCP). The problem (MSCP) consists in minimizing the sum of colors in a graph. Our resolution approach is based on an hybridization of a genetic algorithm and a local heuristic based on an improvement of the maximal independent set algorithm given by F.Glover [4].

Discrete Applied Mathematics

Neighbour-sum-distinguishing edge-weightings are a way to "encode" proper vertex-colourings via the sums of weights incident to the vertices. Over the last decades, this notion has been attracting, in the context of several conjectures, ingrowing attention dedicated, notably, to understanding, which weights are needed to produce neighbour-sum-distinguishing edgeweightings for a given graph. This work is dedicated to investigating another related aspect, namely the minimum number of distinct sums/colours we can produce via a neighbour-sum-distinguishing edgeweighting of a given graph G, and the role of the assigned weights in that context. Clearly, this minimum number is bounded below by the chromatic number χ(G) of G. When using weights of Z, we show that, in general, we can produce neighbour-sum-distinguishing edgeweightings generating χ(G) distinct sums, except in the peculiar case where G is a balanced bipartite graph, in which case χ(G) + 1 distinct sums can be generated. These results are best possible. When using k consecutive weights 1, ..., k, we provide both lower and upper bounds, as a function of the maximum degree ∆, on the maximum least number of sums that can be generated for a graph with maximum degree ∆. For trees, which, in general, admit neighbour-sum-distinguishing 2-edge-weightings, we prove that this maximum, when using weights 1 and 2, is of order 2 log 2 ∆. Finally, we also establish the NP-hardness of several decision problems related to these questions.

Discrete Mathematics, 2010

We define by min c {u,v}∈E(G) |c(u) − c(v)| the min-cost MC (G) of a graph G, where the minimum is taken over all proper colorings c. The min-cost-chromatic number χ M (G) is then defined to be the (smallest) number of colors k for which there exists a proper k-coloring c attaining MC (G). We give constructions of graphs G where χ (G) is arbitrarily smaller than χ M (G). On the other hand, we prove that for every 3-regular graph G , χ M (G) ≤ 4 and for every 4-regular line graph G , χ M (G) ≤ 5. Moreover, we show that the decision problem whether χ M (G) = k is NP-hard for k ≥ 3.

Lecture Notes in Computer Science, 1995

The edge-coloring problem is one of the fundamental problems on graphs, which often appears in various scheduling problems like the file transfer problem on computer networks. In this paper, we survey recent advances and results on the classical edge-coloring problem as well as the generalized edge-coloring problems, called the f-coloring and fg-coloring problems. In particular we review various upper bounds on the minimum number of colors required to edge-color graphs, and present efficient algorithms to edge-color graphs with a number of colors not exceeding the upper bounds.

2011

In this paper we are interested in the elaboration of an approached solution to the sum coloring problem (MSCP), which is an NP-hard problem derived from the graphs coloring (GCP). The problem (MSCP) consists in minimizing the sum of colors in a graph. Our resolution approach is based on an hybridization of a genetic algorithm and a local heuristic based on an improvement of the maximal independent set algorithm given by F.Glover [4].

Journal of Algorithms, 1986

By a result of Holyer, unless P = NP, there does not exist a polynomial-time approximation algorithm to edge color a multigraph that always uses fewer than (f) x' colors, where x' is the optimal number of colors. This makes it appear that finding provably good edge colorings is extremely difficult. However, in this paper we present an algorithm to find an edge coloring of a multigraph that never uses morethan [ix'+tj colors. In addition, if x' 2 1 f A + i] then the algorithm optimal/y colors the graph in polynomial time. Furthermore, this algorithm never uses more than (f)x' colors and runs in O(lEKlVl + A)) time, where E is the set of edges, and P is the set of vertices.

Discrete Applied Mathematics, 2021

A proper coloring of a given graph is an assignment of colors (integer numbers) to its vertices such that two adjacent vertices receives different colors. This paper studies the Minimum Sum Coloring Problem (MSCP), which asks for finding a proper coloring while minimizing the sum of the colors assigned to the vertices. This paper presents the first branch-and-price algorithm to solve the MSCP to proven optimality. The newly developed exact approach is based on an Integer Programming (IP) formulation with an exponential number of variables which is tackled by column generation. Extensive computational experiments, on synthetic and benchmark graphs from the literature, show that the new algorithm outperforms a natural compact IP formulations in terms of: (i) number of solved instances, (ii) running times and (iii) dual gaps obtained when optimality is not achieved. The new exact method is able to solve to optimality 8 out of the 17 (yet unclosed) hard DIMACS instances tested in this work.

Journal of Combinatorial Theory, Series B, 2014

The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb p (G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|, p + 1) colours. In the particular case where G has girth at least p + 1, Arb p (G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring be proper, i.e., adjacent edges receive distinct colours, then the minimum number of colours needed is the generalized p-acyclic edge chromatic number of G. In this paper, we relate the generalized p-acyclic edge chromatic numbers and the generalized p-arboricities of a graph G to the density of the multigraphs having a shallow subdivision as a subgraph of G.

Proceedings of the 13th …, 2008

The volume contains the papers selected for presentation at IPCO 2008, the 13th International Conference on Integer Programming and Combinatorial Optimization that was held in Bertinoro (Italy), May 26-28, 2008.

International Transactions in Operational Research, 2010

This paper surveys the most important algorithmic and computational results on the Vertex Coloring Problem (VCP) and its generalizations. The first part of the paper introduces the classical models for the VCP, and discusses how these models can be used and possibly strengthened to derive exact and heuristic algorithms for the problem. Computational results on the best performing algorithms proposed in the literature are reported. The second part of the paper is devoted to some generalizations of the problem, which are obtained by considering additional constraints [Bandwidth (Multi) Coloring Problem, Bounded Vertex Coloring Problem] or an objective function with a special structure (Weighted Vertex Coloring Problem). The extension of the models for the classical VCP to the considered problems and the best performing algorithms from the literature, as well as the corresponding computational results, are reported.

Discrete Optimization, 2011

Given an undirected graph G = (V , E), the Vertex Coloring Problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. In this paper, we present an exact algorithm for the solution of VCP based on the well-known Set Covering formulation of the problem. We propose a Branch-and-Price algorithm embedding an effective heuristic from the literature and some methods for the solution of the slave problem, as well as two alternative branching schemes. Computational experiments on instances from the literature show the effectiveness of the algorithm, which is able to solve, for the first time to proven optimality, five of the benchmark instances in the literature, and reduce the optimality gap of many others.

AKCE International Journal of Graphs and Combinatorics, 2020

A total coloring of a graph G is an assignment of colors to the vertices and the edges such that (i) no two adjacent vertices receive same color, (ii) no two adjacent edges receive same color, and (iii) if an edge e is incident on a vertex v, then v and e receive different colors. The least number of colors sufficient for a total coloring of graph G is called its total chromatic number and denoted by v 00 ðGÞ: An adjacent vertex distinguishing (AVD)-total coloring of G is a total coloring with the additional property that for any adjacent vertices u and v, the set of colors used on the edges incident on u including the color of u is different from the set of colors used on the edges incident on v including the color of v. The adjacent vertex distinguishing (AVD)-total chromatic number of G, v 00 a ðGÞ is the minimum number of colors required for a valid AVD-total coloring of G. It is conjectured that v 00 ðGÞ DðGÞ þ 2, which is known as total coloring conjecture and is one of the famous open problems. A graph for which the total coloring conjecture holds is called totally colorable graph. The problem of deciding whether v 00 ðGÞ ¼ DðGÞ þ 1 or v 00 ðG ¼ DðGÞ þ 2 for a totally colorable graph G is called the classification problem for total coloring. However, this classification problem is known to be NP-hard even for bipartite graphs. In this paper, we give a sufficient condition for a bipartite biconvex graph G to have v 00 ðGÞ ¼ DðGÞ þ 1: Also, we propose a linear time algorithm to compute the total chromatic number of chain graphs, a proper subclass of biconvex graphs. We prove that the total coloring conjecture holds for the central graph of any graph. Finally, we obtain the AVD-total chromatic number of central graphs for basic graphs such as paths, cycles, stars and complete graphs.

Graphs and Combinatorics, 2019

Preliminaries and Conjectures 2. Tools, Techniques and Classic Results Graph edge coloring has a rich theory, many applications and beautiful conjectures, and it is studied not only by mathematicians, but also by computer scientists. In this survey, written for the nonexpert, we shall describe some main results and techniques and state some of the many popular conjectures in the theory. Besides known results a new basic result about brooms is obtained. Graph edge coloring is a well established subject in the field of graph theory, it is one of the basic combinatorial optimization problems: color the edges of a graph G with as few colors as possible such that each edge receives a color and adjacent edges, that is, different edges incident to a common vertex, receive different colors. The minimum number of colors needed for such a coloring of G is called the chromatic index of G, written χ (G). By a result of Holyer [58], the determination of the chromatic index is an NP-hard optimization problem. The NP-hardness give rise to the necessity of using heuristic algorithms. In particular, we are interested in upper bounds for the chromatic index that can be efficiently realized by a coloring algorithm. A graph parameter ρ is called an efficiently realizable upper bound for χ if there exists an polynomial-time algorithm that colors, for every graph G, the edges of G using at most ρ(G) colors. It follows from a result by Shannon [105] that 3 2 χ is an efficiently realizable upper bound for χ . Motivated by a conjecture proposed, independently, by Goldberg, Andersen, Gupta, and Seymour (see Sect. 1.2), Hochbaum, Nishizeki, and Shmoys [57] made the following suggestion: Conjecture 1 (Hochbaum, Nishizeki, and Shmoys) χ + 1 is an efficiently realizable upper bound for χ .

Anais do Concurso de Teses e Dissertações da SBC (CTD-SBC)

We present a novel recolouring procedure for graph edge-colouring. We show that all graphs whose vertices have local degree sum not too large can be optimally edge-coloured in polynomial time. We also show that the set ofthe graphs satisfying this condition includes almost every graph (under the uniform distribution). We present further results on edge-colouring join graphs, chordal graphs, circular-arc graphs, and complementary prisms, whose proofs yield polynomial-time algorithms. Our results contribute towards settling the Over- full Conjecture, the main open conjecture on edge-colouring simple graphs. Fi- nally, we also present some results on total colouring.