List covering of regular multigraphs

A graph covering projection, also known as a locally bijective homomorphism, is a mapping between vertices and edges of two graphs which preserves incidencies and is a local bijection. This notion stems from topological graph theory, but has also found applications in combinatorics and theoretical computer science. It has been known that for every fixed simple regular graph $H$ of valency greater than 2, deciding if an input graph covers $H$ is NP-complete. In recent years, topological graph theory has developed into heavily relying on multiple edges, loops, and semi-edges, but only partial results on the complexity of covering multigraphs with semi-edges are known so far. In this paper we consider the list version of the problem, called \textsc{List-$H$-Cover}, where the vertices and edges of the input graph come with lists of admissible targets. Our main result reads that the \textsc{List-$H$-Cover} problem is NP-complete for every regular multigraph $H$ of valency greater than 2 which contains at least one semi-simple vertex (i.e., a vertex which is incident with no loops, with no multiple edges and with at most one semi-edge). Using this result we almost show the NP-co/polytime dichotomy for the computational complexity of \textsc{ List-$H$-Cover} of cubic multigraphs, leaving just five open cases.Comment: Accepted to IWOCA 202

Novel procedures for graph edge-colouring

Orientador: Dr. Renato CarmoCoorientador: Dr. André Luiz Pires GuedesTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa : Curitiba, 05/12/2018Inclui referências e índiceÁrea de concentração: Ciência da ComputaçãoResumo: O índice cromático de um grafo G é o menor número de cores necessário para colorir as arestas de G de modo que não haja duas arestas adjacentes recebendo a mesma cor. Pelo célebre Teorema de Vizing, o índice cromático de qualquer grafo simples G ou é seu grau máximo , ou é ? + 1, em cujo caso G é dito Classe 1 ou Classe 2, respectivamente. Computar uma coloração de arestas ótima de um grafo ou simplesmente determinar seu índice cromático são problemas NP-difíceis importantes que aparecem em aplicações notáveis, como redes de sensores, redes ópticas, controle de produção, e jogos. Neste trabalho, nós apresentamos novos procedimentos de tempo polinomial para colorir otimamente as arestas de grafos pertences a alguns conjuntos grandes. Por exemplo, seja X a classe dos grafos cujos maiorais (vértices de grau ?) possuem soma local de graus no máximo ?2 ?? (entendemos por 'soma local de graus' de um vértice x a soma dos graus dos vizinhos de x). Nós mostramos que quase todo grafo está em X e, estendendo o procedimento de recoloração que Vizing usou na prova para seu teorema, mostramos que todo grafo em X é Classe 1. Nós também conseguimos resultados em outras classes de grafos, como os grafos-junção, os grafos arco-circulares, e os prismas complementares. Como um exemplo, nós mostramos que um prisma complementar só pode ser Classe 2 se for um grafo regular distinto do K2. No que diz respeito aos grafos-junção, nós mostramos que se G1 e G2 são grafos disjuntos tais que |V(G1)| _ |V(G2)| e ?(G1) _ ?(G2), e se os maiorais de G1 induzem um grafo acíclico, então o grafo-junção G1 ?G2 é Classe 1. Além desses resultados em coloração de arestas, apresentamos resultados parciais em coloração total de grafos-junção, de grafos arco-circulares, e de grafos cobipartidos, bem como discutimos um procedimento de recoloração para coloração total. Palavras-chave: Coloração de grafos e hipergrafos (MSC 05C15). Algoritmos de grafos (MSC 05C85). Teoria dos grafos em relação à Ciência da Computação (MSC 68R10). Graus de vértices (MSC 05C07). Operações de grafos (MSC 05C76).Abstract: The chromatic index of a graph G is the minimum number of colours needed to colour the edges of G in a manner that no two adjacent edges receive the same colour. By the celebrated Vizing's Theorem, the chromatic index of any simple graph G is either its maximum degree ? or it is ? + 1, in which case G is said to be Class 1 or Class 2, respectively. Computing an optimal edge-colouring of a graph or simply determining its chromatic index are important NP-hard problems which appear in noteworthy applications, like sensor networks, optical networks, production control, and games. In this work we present novel polynomial-time procedures for optimally edge-colouring graphs belonging to some large sets of graphs. For example, let X be the class of the graphs whose majors (vertices of degree ?) have local degree sum at most ?2 ? ? (by 'local degree sum' of a vertex x we mean the sum of the degrees of the neighbours of x). We show that almost every graph is in X and, by extending the recolouring procedure used by Vizing's in the proof for his theorem, we show that every graph in X is Class 1. We further achieve results in other graph classes, such as join graphs, circular-arc graphs, and complementary prisms. For instance, we show that a complementary prism can be Class 2 only if it is a regular graph distinct from the K2. Concerning join graphs, we show that if G1 and G2 are disjoint graphs such that |V(G1)| _ |V(G2)| and ?(G1) _ ?(G2), and if the majors of G1 induce an acyclic graph, then the join graph G1 ?G2 is Class 1. Besides these results on edge-colouring, we present partial results on total colouring join graphs, cobipartite graphs, and circular-arc graphs, as well as a discussion on a recolouring procedure for total colouring. Keywords: Colouring of graphs and hypergraphs (MSC 05C15). Graph algorithms (MSC 05C85). Graph theory in relation to Computer Science (MSC 68R10). Vertex degrees (MSC 05C07). Graph operations (MSC 05C76)

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM