Minimum Sum Edge Colorings of Multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {\em chromatic edge strength} of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems

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

Coloración en triangulaciones

Some of the most studied problems in Graph Theory are those referring to the coloring of the graph, being one of the most famous the Three Color Problem. A color set D for a graph G is said to be a 3-coloring if adjacent vertex has a different color of D making the graph 3-coloreable. It seems to be obvious to wonder which graphs are 3-coloreable. Nevertheless, the problem of finding sufficient conditions for a graph to be 3-coloreable in a general graph has been shown by L. Stockmayer in 1979 in his book “Planar 3-colorability is polynomial complete" to be NP-complete. That is why different bounds for x(G) are studied and stated for both arbitrary graphs and for those with a particular structure. Nonetheless, the interest in this parameter is not only to establish new bounds, but also once the bounds have been obtained, either upper or bottom, this naturally brings us the question of knowing if there exists any graph which verifies the equality. Throughout these months, the results achieved about the 3-coloring problem for arbitrary graphs have been studied and, specifically, those results referring to the variants of the 3-coloration problem attending to the sum of colors, the distance between vertex or the parity among the apparition of certain color. This research has been performed not only from a combinatorial point of view but also from an algorithmic point of view and has been restricted to a particular kind of graph, known as maximal outerplanar graphs and denoted by its acronym as MOP's, graph of high importance in both the field of chemistry and polygon triangulations. This project has a double purpose: on the one hand, it seeks to collect those results in the literature which have been observed to be more significant in a review paper or sur- vey; on the other hand, it seeks to established tight combinatorial bounds for some variants of the 3-coloration concept for any n-vertex maximal outerplanar graph. Thus, as main contributions, we will prove several new tight combinatorial bounds for the following variants of coloration concept attending to the sum of the colors been used: sum-coloring, as well as the following variants attending to the existence of a rainbow path: rainbow coloring.---ABSTRACT---Algunos de los problemas más estudiados en Teoría de Grafos son aquellos problemas que hacen referencia a la coloración del mismo, siendo uno de los más clásicos el problema de los Tres Colores. Un conjunto D de colores de un grafo G se dice que es una 3-coloración si vértices adyacentes tienen un color distinto de D haciendo el grafo 3-coloreable. Parece entonces obvio preguntarse qué grafos son 3-coloreables. Sin embargo, ya en 1979 L. Stockmayer en su artículo “Planar 3-colorability is polynomial complete" probó que este problema es NP-completo. Es por ello por lo que se estudian y establecen cotas para x(G) para el caso de grafos cualesquiera o para grafos con cierta estructura. Sin embargo, el interés en este parámetro no sólo radica en establecer una cota, sino que una vez obtenida dicha cota, ya sea superior o inferior, quedaría comprobar la existencia de algún grafo que verifique la igualdad. A lo largo de estos meses de trabajo, se han estudiado los resultados obtenidos hasta la fecha en el problema de la 3-coloración de grafos en general y más concretamente sobre aquellas variantes de 3-coloración que atienden a la suma de los colores, la distancia entre vértices o la paridad en la aparición de cierto color. Este estudio se ha llevado a cabo tanto desde el punto de vista combinatorio como algorítmico y se ha restringido a un tipo particular de grafos, conocidos como grafos periplanos maximales y de nominados a partir de ahora por sus siglas en inglés MOP's (maximal outerplanar graphs), grafos de gran importancia tanto en el ámbito de la química como en el de triangulaciones de polígonos. Con este proyecto se persigue un doble objetivo: por un lado, se pretende recopilar aque- llos resultados más significativos de la bibliografía en un artículo de tipo "survey"; por otro, obtener nuevos resultados sobre variantes de dominación para MOP's. Así, como aporte de nuestro trabajo, probaremos nuevas cotas que se han establecido tanto para los criterios de 3-coloración que atienden a la suma de colores utilizados en la coloración: sum-coloring, como para variantes que atienden a la existencia de caminos irisados en la coloración del grafo: coloración irisada