12,327 research outputs found
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
Random runners are very lonely
Suppose that runners having different constant speeds run laps on a
circular track of unit length. The Lonely Runner Conjecture states that, sooner
or later, any given runner will be at distance at least from all the
other runners. We prove that, with probability tending to one, a much stronger
statement holds for random sets in which the bound is replaced by
\thinspace . The proof uses Fourier analytic methods. We also
point out some consequences of our result for colouring of random integer
distance graphs
Colouring random graphs and maximising local diversity
We study a variation of the graph colouring problem on random graphs of
finite average connectivity. Given the number of colours, we aim to maximise
the number of different colours at neighbouring vertices (i.e. one edge
distance) of any vertex. Two efficient algorithms, belief propagation and
Walksat are adapted to carry out this task. We present experimental results
based on two types of random graphs for different system sizes and identify the
critical value of the connectivity for the algorithms to find a perfect
solution. The problem and the suggested algorithms have practical relevance
since various applications, such as distributed storage, can be mapped onto
this problem.Comment: 10 pages, 10 figure
The distance-t chromatic index of graphs
We consider two graph colouring problems in which edges at distance at most
are given distinct colours, for some fixed positive integer . We obtain
two upper bounds for the distance- chromatic index, the least number of
colours necessary for such a colouring. One is a bound of (2-\eps)\Delta^t
for graphs of maximum degree at most , where \eps is some absolute
positive constant independent of . The other is a bound of (as ) for graphs of maximum degree at most
and girth at least . The first bound is an analogue of Molloy and Reed's
bound on the strong chromatic index. The second bound is tight up to a constant
multiplicative factor, as certified by a class of graphs of girth at least ,
for every fixed , of arbitrarily large maximum degree , with
distance- chromatic index at least .Comment: 14 pages, 2 figures; to appear in Combinatorics, Probability and
Computin
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
Planar graphs without 3-cycles and with 4-cycles far apart are 3-choosable
A graph G is said to be L-colourable if for a given list assignment L = {L(v)|v ∈ V (G)} there is a proper colouring c of G such that c(v) ∈ L(v) for all v in V (G). If G is L-colourable for all L with |L(v)| ≥ k for all v in V (G), then G is said to be k-choosable.
This paper focuses on two different ways to prove list colouring results on planar graphs. The first method will be discharging, which will be used to fuse multiple results into one theorem. The second method will be restricting the lists of vertices on the boundary and applying induction, which will show that planar graphs without 3- cycles and 4-cycles distance 8 apart are 3-choosable
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