Algorithms and Bounds for Very Strong Rainbow Coloring

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in $k$ and the treewidth of $G$. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor $n^{1-\varepsilon}$, unless P$=$NP

Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study

Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds

We study rainbow connectivity of graphs from the algorithmic and graph-theoretic points of view. The study is divided into three parts. First, we study the complexity of deciding whether a given edge-colored graph is rainbow-connected. That is, we seek to verify whether the graph has a path on which no color repeats between each pair of its vertices. We obtain a comprehensive map of the hardness landscape of the problem. While the problem is NP-complete in general, we identify several structural properties that render the problem tractable. At the same time, we strengthen the known NP-completeness results for the problem. We pinpoint various parameters for which the problem is fixed-parameter tractable, including dichotomy results for popular width parameters, such as treewidth and pathwidth. The study extends to variants of the problem that consider vertex-colored graphs and/or rainbow shortest paths. We also consider upper and lower bounds for exact parameterized algorithms. In particular, we show that when parameterized by the number of colors k, the existence of a rainbow s-t path can be decided in O∗ (2k ) time and polynomial space. For the highly related problem of finding a path on which all the k colors appear, i.e., a colorful path, we explain the modest progress over the last twenty years. Namely, we prove that the existence of an algorithm for finding a colorful path in (2 − ε)k nO(1) time for some ε > 0 disproves the so-called Set Cover Conjecture.Second, we focus on the problem of finding a rainbow coloring. The minimum number of colors for which a graph G is rainbow-connected is known as its rainbow connection number, denoted by rc(G). Likewise, the minimum number of colors required to establish a rainbow shortest path between each pair of vertices in G is known as its strong rainbow connection number, denoted by src(G). We give new hardness results for computing rc(G) and src(G), including their vertex variants. The hardness results exclude polynomial-time algorithms for restricted graph classes and also fast exact exponential-time algorithms (under reasonable complexity assumptions). For positive results, we show that rainbow coloring is tractable for e.g., graphs of bounded treewidth. In addition, we give positive parameterized results for certain variants and relaxations of the problems in which the goal is to save k colors from a trivial upper bound, or to rainbow connect only a certain number of vertex pairs.Third, we take a more graph-theoretic view on rainbow coloring. We observe upper bounds on the rainbow connection numbers in terms of other well-known graph parameters. Furthermore, despite the interest, there have been few results on the strong rainbow connection number of a graph. We give improved bounds and determine exactly the rainbow and strong rainbow connection numbers for some subclasses of chordal graphs. Finally, we pose open problems and conjectures arising from our work

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