2,569 research outputs found
Asymmetric coloring games on incomparability graphs
Consider the following game on a graph : Alice and Bob take turns coloring
the vertices of 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 is the minimum number of colors
that allows Alice to win the game. The game Grundy number of 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 -game
chromatic and Grundy numbers are defined likewise except that Alice colors
vertices and Bob colors 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 for which the
-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
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
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
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
Fine-Grained Complexity of Rainbow Coloring and its Variants
Consider a graph G and an edge-coloring c_R:E(G) rightarrow [k]. A rainbow path between u,v in V(G) is a path P from u to v such that for all e,e\u27 in E(P), where e neq e\u27 we have c_R(e) neq c_R(e\u27). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all u,v in V(G) there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S subseteq V(G) times V(G), and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all (u,v) in S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S subseteq V(G), and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all u,v in S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results.
- For every k geq 3, Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|E(G)|)}n^{O(1)}, unless ETH fails.
- For every k geq 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|S|^2)}n^{O(1)}, unless ETH fails.
- Subset Rainbow k-Coloring admits an algorithm running in time 2^{OO(|S|)}n^{O(1)}. This also implies an algorithm running in time 2^{o(|S|^2)}n^{O(1)} for Steiner Rainbow k-Coloring, which matches the lower bound we obtain
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