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Edge and total choosability of near-outerplanar graphs
It is proved that, if G is a K4-minor-free graph with maximum degree ∆ ≥ 4, then G is totally (∆ + 1)-choosable; that is, if every element (vertex or edge) of G is assigned a list of ∆ + 1 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ch’’(G) = χ’(G) for every graph G, is true for all K4-minor-free graphs. The List-Edge-Colouring Conjecture is also known to be true for these graphs. As a fairly straightforward consequence, it is proved that both conjectures hold also for all K2,3-minor free graphs and all ( K2 + (K1 U K2))-minor-free graphs
A note on total and list edge-colouring of graphs of tree-width 3
It is shown that Halin graphs are -edge-choosable and that graphs of
tree-width 3 are -edge-choosable and -total-colourable.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0212
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
The Semantics of Graph Programs
GP (for Graph Programs) is a rule-based, nondeterministic programming
language for solving graph problems at a high level of abstraction, freeing
programmers from handling low-level data structures. The core of GP consists of
four constructs: single-step application of a set of conditional
graph-transformation rules, sequential composition, branching and iteration. We
present a formal semantics for GP in the style of structural operational
semantics. A special feature of our semantics is the use of finitely failing
programs to define GP's powerful branching and iteration commands
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
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