5 research outputs found

    Graphs and subgraphs with bounded degree

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    "The topology of a network (such as a telecommunications, multiprocessor, or local area network, to name just a few) is usually modelled by a graph in which vertices represent 'nodes' (stations or processors) while undirected or directed edges stand for 'links' or other types of connections, physical or virtual. A cycle that contains every vertex of a graph is called a hamiltonian cycle and a graph which contains a hamiltonian cycle is called a hamiltonian graph. The problem of the existence of a hamiltonian cycle is closely related to the well known problem of a travelling salesman. These problems are NP-complete and NP-hard, respectively. While some necessary and sufficient conditions are known, to date, no practical characterization of hamiltonian graphs has been found. There are several ways to generalize the notion of a hamiltonian cycle. In this thesis we make original contributions in two of them, namely k-walks and r-trestles." --Abstract.Doctor of Philosoph

    Toughness threshold for the existence of 2-walks in K4-minor free graphs

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    We show that every K4-minor free graph with toughness greater than 4/7 has a 2-walk, i.e., a closed walk visiting each vertex at most twice. We also give an example of a 4/7-tough K4-minor free graph with no 2-walk. 1 Introduction An active area of graph theory is the study of Hamilton cycles [8, 9], in par-ticular, the study of conditions based on different connectivity parameters that guarantees the existence of a Hamilton cycle in a graph. One of themost famous conjectures in this area is Chv'atal's conjecture. Its original version asserts that every 2-tough graph G is hamiltonian. Let us recallthat a graph G is hamiltonian if it contains a cycle passing through all itsvertices, and G is ff-tough if the number o / (A) of components of G \ A is atmost max{1, |A|/ff} for every non-empty set A of the vertices. The originalconjecture has been disproved by Bauer et al. [1] who constructed (

    Generalized Colorings of Graphs

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    A graph coloring is an assignment of labels called ā€œcolorsā€ 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 āˆ’ 1), conļ¬rming 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 deļ¬ne 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 diļ¬€erent 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
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