878 research outputs found
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 -vertex graph with bounded degeneracy has a three-dimensional grid
drawing with 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
volume, which is tight for the complete dag. The previous best upper
bound was . Our main result is that every -colourable directed
acyclic graph ( constant) has an upward three-dimensional grid drawing with
volume. This result matches the bound in the undirected case, and
improves the best known bound from for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar
Complete Acyclic Colorings
We study two parameters that arise from the dichromatic number and the
vertex-arboricity in the same way that the achromatic number comes from the
chromatic number. The adichromatic number of a digraph is the largest number of
colors its vertices can be colored with such that every color induces an
acyclic subdigraph but merging any two colors yields a monochromatic directed
cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest
number of colors that can be used such that every color induces a forest but
merging any two yields a monochromatic cycle. We study the relation between
these parameters and their behavior with respect to other classical parameters
such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure
On the oriented chromatic number of dense graphs
Let be a graph with vertices, edges, average degree , and maximum degree . The \emph{oriented chromatic number} of is the maximum, taken over all orientations of , of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which . We prove that every such graph has oriented chromatic number at least . In the case that , this lower bound is improved to . Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when is ()-regular for some constant , in which case the oriented chromatic number is between and
Connection Matrices and the Definability of Graph Parameters
In this paper we extend the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with modular counting CMSOL of B. Godlin, T. Kotek and J.A. Makowsky (2008 and 2009), and demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers
Pewarnaan harmonis pada beberapa kelas graf berarah
AbstrakPewarnaan graf merupakan suatu pemetaan dari elemen pada suatu graf ke himpunan semua bilangan asli sedemikian sehingga setiap elemen yang bertetangga tidak dipetakan ke bilangan yang sama. Pada pewarnaan graf, image dari elemen suatu graf disebut warna. Dimisalkan dan adalah simpul-simpul pada dan serta adalah warna. Jika simpul diwarnai dengan dan simpul diwarnai dengan maka pasangan warna yang dihasilkan adalah pasangan warna . Pewarnaan harmonis menerapkan konsep pewarnaan simpul dalam mewarnai suatu graf dengan syarat satu pasang warna muncul paling banyak satu kali. Banyak warna yang paling minimum yang digunakan dalam pewarnaan harmonis disebut bilangan kromatik harmonis. Dalam penelitian ini, konsep pewarnaan harmonis akan diterapkan pada beberapa kelas graf berarah untuk melihat pola bilangan kromatik dari masing-masing kelas tersebut. Seperti diketahui, pada graf berarah , pasangan warna sehingga proses pewarnaan tersebut menjadi lebih kompleks. Adapun kelas graf yang dibahas adalah graf lili berarah , graf komplit berarah dan graf kipas berarah . Didapat bilangan kromatik harmonis pada graf lili berarah adalah dengan ; bilangan kromatik harmonis pada graf komplit berarah adalah . Sedangkan bilangan kromatik harmonis pada pewarnaan graf kipas berarah berada pada selang .Kata kunci: bilangan kromatik harmonis; pasangan warna; pewarnaan simpu
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