157 research outputs found
Track Layouts of Graphs
A \emph{-track layout} of a graph consists of a (proper) vertex
-colouring of , a total order of each vertex colour class, and a
(non-proper) edge -colouring such that between each pair of colour classes
no two monochromatic edges cross. This structure has recently arisen in the
study of three-dimensional graph drawings. This paper presents the beginnings
of a theory of track layouts. First we determine the maximum number of edges in
a -track layout, and show how to colour the edges given fixed linear
orderings of the vertex colour classes. We then describe methods for the
manipulation of track layouts. For example, we show how to decrease the number
of edge colours in a track layout at the expense of increasing the number of
tracks, and vice versa. We then study the relationship between track layouts
and other models of graph layout, namely stack and queue layouts, and geometric
thickness. One of our principle results is that the queue-number and
track-number of a graph are tied, in the sense that one is bounded by a
function of the other. As corollaries we prove that acyclic chromatic number is
bounded by both queue-number and stack-number. Finally we consider track
layouts of planar graphs. While it is an open problem whether planar graphs
have bounded track-number, we prove bounds on the track-number of outerplanar
graphs, and give the best known lower bound on the track-number of planar
graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as
Technical Report TR-2003-07, School of Computer Science, Carleton University,
Ottawa, Canad
The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs
The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k
Colorings of oriented planar graphs avoiding a monochromatic subgraph
For a fixed simple digraph and a given simple digraph , an -free
-coloring of is a vertex-coloring in which no induced copy of in
is monochromatic. We study the complexity of deciding for fixed and
whether a given simple digraph admits an -free -coloring. Our main focus
is on the restriction of the problem to planar input digraphs, where it is only
interesting to study the cases . From known results it follows
that for every fixed digraph whose underlying graph is not a forest, every
planar digraph admits an -free -coloring, and that for every fixed
digraph with , every oriented planar graph admits an
-free -coloring.
We show in contrast, that
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring.
- if is an orientation of a path of length at least , then it is
NP-hard to decide whether an acyclic and planar input digraph admits an
-free -coloring
- …