Harmonious Coloring of Trees with Large Maximum Degree

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a harmonious coloring of $G$. We show that if $T$ is a forest of order $n$ with maximum degree $\Delta(T)\geq \frac{n+2}{3}$, then h(T)= \Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$; \Delta(T)+1, & otherwise. Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.Comment: 8 pages, 1 figur

Upward Three-Dimensional Grid Drawings of Graphs

A \emph{three-dimensional grid drawing} of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. We prove that every $n$-vertex graph with bounded degeneracy has a three-dimensional grid drawing with $O(n^{3/2})$ volume. This is the broadest class of graphs admiting such drawings. A three-dimensional grid drawing of a directed graph is \emph{upward} if every arc points up in the z-direction. We prove that every directed acyclic graph has an upward three-dimensional grid drawing with $(n^3)$ volume, which is tight for the complete dag. The previous best upper bound was $O(n^4)$. Our main result is that every $c$-colourable directed acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with $O(n^2)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $O(n^3)$ for many classes of directed acyclic graphs, including planar, series parallel, and outerplanar

A note on "Folding wheels and fans."

In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique numbers onto which wheels and fans fold. We present an interpolation theorem which generalizes their theorems 4.2 and 5.2. We show that their formula for wheels is wrong. We show that for threshold graphs, the achromatic number and folding number coincides with the chromatic number

Asymmetric coloring games on incomparability graphs

Consider the following game on a graph $G$: Alice and Bob take turns coloring the vertices of $G$ properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of $G$ is the minimum number of colors that allows Alice to win the game. The game Grundy number of $G$ is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The $(a,b)$-game chromatic and Grundy numbers are defined likewise except that Alice colors $a$ vertices and Bob colors $b$ vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs $(a,b)$ for which the $(a,b)$-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu

Rotulações próprias por gap : variantes de arestas e de vértices

Orientadores: Christiane Neme Campos, Rafael Crivellari Saliba SchoueryDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Uma rotulação própria é uma atribuição de valores numéricos aos elementos de um grafo, que podem ser seus vértices, arestas ou ambos, de modo a obter - usando certas funções matemáticas sobre o conjunto de rótulos - uma coloração dos vértices do grafo tal que nenhum par de vértices adjacentes receba a mesma cor. Este texto aborda o problema da rotulação própria por gap em suas versões de arestas e de vértices. Na versão de arestas, um vértice de grau pelo menos dois tem sua cor definida como a maior diferença, i.e. o maior gap, entre os rótulos de suas arestas incidentes; já na variante de vértices, o gap é definido pela maior diferença entre os rótulos dos seus vértices adjacentes. Para vértices de grau um, sua cor é dada pelo rótulo da sua aresta incidente, no caso da versão de arestas, e pelo rótulo de seu vértice adjacente, no caso da versão de vértices. Finalmente, vértices de grau zero recebem cor um. O menor número de rótulos para o qual um grafo admite uma rotulação própria por gap de arestas vértices é chamado edge-gap (vertex-gap) number. Neste trabalho, apresentamos um breve histórico das rotulações próprias por gap e os resultados obtidos para as duas versões do problema. Estudamos o edge-gap e o vertex-gap numbers para as famílias de ciclos, coroas, rodas, grafos unicíclicos e algumas classes de snarks. Adicionalmente, na versão de vértices, investigamos a família de grafos cúbicos bipartidos hamiltonianos, desenvolvendo diversas técnicas de rotulação para grafos nesta classe. Em uma abordagem relacionada, provamos resultados de complexidade para a família dos grafos subcúbicos bipartidos. Além disso, demonstramos propriedades estruturais destas rotulações de vértices por gap e estabelecemos limitantes inferiores e superiores justos para o vertex-gap number de grafos arbitrários. Mostramos, ainda, que nem todos os grafos admitem uma rotulação de vértices por gap, exibindo duas famílias infinitas que não admitem tal rotulação. A partir dessas classes, definimos um novo parâmetro chamado de gap-strength, referente ao menor número de arestas que precisam ser removidas de um grafo de modo a obter um novo grafo que é gap-vértice-rotulável. Estabelecemos um limitante superior para o gap-strength de grafos completos e apresentamos evidências de que este valor pode ser um limitante inferiorAbstract: A proper labelling is an assignment of numerical values to the elements of a graph, which can be vertices, edges or both, so as to obtain - through the use of mathematical functions over the set of labels - a vertex-colouring of the graph such that every pair of adjacent vertices receives different colours. This text addresses the proper gap-labelling problem in its edge and vertex variants. In the former, a vertex of degree at least two has its colour defined by the largest difference, or gap, among the labels of its incident edges; in the vertex variant, the gap is defined by the largest difference among the labels of its adjacent vertices. For a degree-one vertex, its colour is defined by the label of its incident edge, in the edge version, and by the label of its adjacent vertex, in the vertex variant. Finally, degree-zero vertices receive colour one. The least number of labels for which a graph admits a proper gap-labelling of its edges (vertices) is called the edge-gap (vertex-gap) number. In this work, we present a brief history of proper gap-labellings and our results for both versions of the problem. We study the edge-gap and vertex-gap numbers for the families of cycles, crowns, wheels, unicyclic graphs and some classes of snarks. Additionally, in the vertex version, we investigate the family of cubic bipartite hamiltonian graphs and develop several labelling techniques for graphs in this class. In a related approach, we prove hardness results for the family of subcubic bipartite graphs. Also, we demonstrate structural properties of gap-vertex-labelable graphs and establish tight lower and upper bounds for the vertex-gap number of arbitrary graphs. We also show that not all graphs admit a proper gap-labelling, exhibiting two infinite families of graphs for which no such vertex-labelling exists. Thus, we define a new parameter called the gap-strength of graphs, which is the least number of edges that have to be removed from a graph so as to obtain a new, gap-vertex-labelable graph. We establish an upper bound for the gap-strength of complete graphs and argue that this value can also be used as a lower boundMestradoCiência da ComputaçãoMestre em Ciência da ComputaçãoCAPE

The Parameterized Complexity of Degree Constrained Editing Problems

This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are within the given constraints. Parameterized complexity gives a framework for examining problems that are traditionally considered intractable and developing efficient exact algorithms for them, or showing that it is unlikely that they have such algorithms, by introducing an additional component to the input, the parameter, which gives additional information about the structure of the problem. If the problem has an algorithm that is exponential in the parameter, but polynomial, with constant degree, in the size of the input, then it is considered to be fixed-parameter tractable. Parameterized complexity also provides an intractability framework for identifying problems that are likely to not have such an algorithm. Degree constrained editing problems provide natural parameterizations in terms of the total cost k of vertex deletions, edge deletions and edge additions allowed, and the upper bound r on the degree of the vertices remaining after editing. We define a class of degree constrained editing problems, WDCE, which generalises several well know problems, such as Degree r Deletion, Cubic Subgraph, r-Regular Subgraph, f-Factor and General Factor. We show that in general if both k and r are part of the parameter, problems in the WDCE class are fixed-parameter tractable, and if parameterized by k or r alone, the problems are intractable in a parameterized sense. We further show cases of WDCE that have polynomial time kernelizations, and in particular when all the degree constraints are a single number and the editing operations include vertex deletion and edge deletion we show that there is a kernel with at most O(kr(k + r)) vertices. If we allow vertex deletion and edge addition, we show that despite remaining fixed-parameter tractable when parameterized by k and r together, the problems are unlikely to have polynomial sized kernelizations, or polynomial time kernelizations of a certain form, under certain complexity theoretic assumptions. We also examine a more general case where given an input graph the question is whether with at most k deletions the graph can be made r-degenerate. We show that in this case the problems are intractable, even when r is a constant