175 research outputs found
On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor
The generalised colouring numbers and
were introduced by Kierstead and Yang as a generalisation
of the usual colouring number, and have since then found important theoretical
and algorithmic applications. In this paper, we dramatically improve upon the
known upper bounds for generalised colouring numbers for graphs excluding a
fixed minor, from the exponential bounds of Grohe et al. to a linear bound for
the -colouring number and a polynomial bound for the weak
-colouring number . In particular, we show that if
excludes as a minor, for some fixed , then
and
.
In the case of graphs of bounded genus , we improve the bounds to
(and even if
, i.e. if is planar) and
.Comment: 21 pages, to appear in European Journal of Combinatoric
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
On the generalised colouring numbers of graphs that exclude a fixed minor
The generalised colouring numbers colr(G) and wcolr(G) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r-colouring number colr and a polynomial bound for the weak r-colouring number wcolr. In particular, we show that if G excludes Kt as a minor, for some fixed tβ₯4, then colr(G)β€(tβ12)(2r+1) and wcolr(G)β€(r+tβ2tβ2)β
(tβ3)(2r+1)βO(rtβ1). In the case of graphs G of bounded genus g, we improve the bounds to colr(G)β€(2g+3)(2r+1) (and even colr(G)β€5r+1 if g=0, i.e. if G is planar) and wcolr(G)β€(2g+(r+22))(2r+1)
Acyclic edge coloring of subcubic graphs
AbstractAn acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by aβ²(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs
Bounded expansion and nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of uniformly sparse graphs which includes the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs. Since their initial definition it was shown that these graph classes can be defined in many equivalent ways: by generalised colouring numbers, neighbourhood complexity, sparse neighbourhood covers, a game known as the splitter game, and many more.
We study the corresponding concepts for directed graphs. We show that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case. We provide a characterisation of bounded expansion classes by a directed version of the generalised colouring numbers. As an application we show how to construct sparse directed neighbourhood covers and how to approximate directed distance-r dominating sets on classes of bounded expansion. On the other hand, we show that linear neighbourhood complexity does not characterise directed classes of bounded expansion
- β¦