164 research outputs found
New Bounds for Facial Nonrepetitive Colouring
We prove that the facial nonrepetitive chromatic number of any outerplanar
graph is at most 11 and of any planar graph is at most 22.Comment: 16 pages, 5 figure
Nonrepetitive Colourings of Planar Graphs with Colours
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for
which the first half of the path is assigned the same sequence of colours as
the second half. The \emph{nonrepetitive chromatic number} of a graph is
the minimum integer such that has a nonrepetitive -colouring.
Whether planar graphs have bounded nonrepetitive chromatic number is one of the
most important open problems in the field. Despite this, the best known upper
bound is for -vertex planar graphs. We prove a
upper bound
Layout of Graphs with Bounded Tree-Width
A \emph{queue layout} of a graph consists of a total order of the vertices,
and a partition of the edges into \emph{queues}, such that no two edges in the
same queue are nested. The minimum number of queues in a queue layout of a
graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line
grid) drawing} of a graph represents the vertices by points in
and the edges by non-crossing line-segments. This paper contributes three main
results:
(1) It is proved that the minimum volume of a certain type of
three-dimensional drawing of a graph is closely related to the queue-number
of . In particular, if is an -vertex member of a proper minor-closed
family of graphs (such as a planar graph), then has a drawing if and only if has O(1) queue-number.
(2) It is proved that queue-number is bounded by tree-width, thus resolving
an open problem due to Ganley and Heath (2001), and disproving a conjecture of
Pemmaraju (1992). This result provides renewed hope for the positive resolution
of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional
drawings with O(n) volume. This is the most general family of graphs known to
admit three-dimensional drawings with O(n) volume.
The proofs depend upon our results regarding \emph{track layouts} and
\emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October
2002. This paper incorporates the following conference papers: (1) Dujmovic',
Morin & Wood. Path-width and three-dimensional straight-line grid drawings of
graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts,
tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS
2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of
-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200
Rectilinear Crossing Number of Graphs Excluding Single-Crossing Graphs as Minors
The crossing number of a graph is the minimum number of crossings in a
drawing of in the plane. A rectilinear drawing of a graph represents
vertices of by a set of points in the plane and represents each edge of
by a straight-line segment connecting its two endpoints. The rectilinear
crossing number of is the minimum number of crossings in a rectilinear
drawing of .
By the crossing lemma, the crossing number of an -vertex graph can be
only if . Graphs of bounded genus and bounded degree
(B\"{o}r\"{o}czky, Pach and T\'{o}th, 2006) and in fact all bounded degree
proper minor-closed families (Wood and Telle, 2007) have been shown to admit
linear crossing number, with tight bound shown by
Dujmovi\'c, Kawarabayashi, Mohar and Wood, 2008.
Much less is known about rectilinear crossing number. It is not bounded by
any function of the crossing number. We prove that graphs that exclude a
single-crossing graph as a minor have the rectilinear crossing number . This dependence on and is best possible. A single-crossing
graph is a graph whose crossing number is at most one. Thus the result applies
to -minor-free graphs, for example. It also applies to bounded treewidth
graphs, since each family of bounded treewidth graphs excludes some fixed
planar graph as a minor. Prior to our work, the only bounded degree
minor-closed families known to have linear rectilinear crossing number were
bounded degree graphs of bounded treewidth (Wood and Telle, 2007), as well as,
bounded degree -minor-free graphs (Dujmovi\'c, Kawarabayashi, Mohar
and Wood, 2008). In the case of bounded treewidth graphs, our
result is again tight and improves on the previous best known bound of
by Wood and Telle, 2007 (obtained for convex geometric
drawings)
Anagram-Free Chromatic Number is not Pathwidth-Bounded
The anagram-free chromatic number is a new graph parameter introduced
independently Kam\v{c}ev, {\L}uczak, and Sudakov (2017) and Wilson and Wood
(2017). In this note, we show that there are planar graphs of pathwidth 3 with
arbitrarily large anagram-free chromatic number. More specifically, we describe
-vertex planar graphs of pathwidth 3 with anagram-free chromatic number
. We also describe vertex graphs with pathwidth
having anagram-free chromatic number in .Comment: 8 pages, 3 figure
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