8,481 research outputs found
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Playing with parameters: structural parameterization in graphs
When considering a graph problem from a parameterized point of view, the
parameter chosen is often the size of an optimal solution of this problem (the
"standard" parameter). A natural subject for investigation is what happens when
we parameterize such a problem by various other parameters, some of which may
be the values of optimal solutions to different problems. Such research is
known as parameterized ecology. In this paper, we investigate seven natural
vertex problems, along with their respective parameters: the size of a maximum
independent set, the size of a minimum vertex cover, the size of a maximum
clique, the chromatic number, the size of a minimum dominating set, the size of
a minimum independent dominating set and the size of a minimum feedback vertex
set. We study the parameterized complexity of each of these problems with
respect to the standard parameter of the others.Comment: 17 page
On-line list colouring of random graphs
In this paper, the on-line list colouring of binomial random graphs G(n,p) is
studied. We show that the on-line choice number of G(n,p) is asymptotically
almost surely asymptotic to the chromatic number of G(n,p), provided that the
average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log
n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that
if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is
larger than the chromatic number by at most a multiplicative factor of C, where
C in [2,4], depending on the range of d. Also, for d=O(1), the on-line choice
number is by at most a multiplicative constant factor larger than the chromatic
number
Progress on the adjacent vertex distinguishing edge colouring conjecture
A proper edge colouring of a graph is adjacent vertex distinguishing if no
two adjacent vertices see the same set of colours. Using a clever application
of the Local Lemma, Hatami (2005) proved that every graph with maximum degree
and no isolated edge has an adjacent vertex distinguishing edge
colouring with colours, provided is large enough. We
show that this bound can be reduced to . This is motivated by the
conjecture of Zhang, Liu, and Wang (2002) that colours are enough
for .Comment: v2: Revised following referees' comment
Graph colouring for office blocks
The increasing prevalence of WLAN (wireless networks) introduces the potential of electronic information leakage from one company's territory in an office block, to others due to the long-ranged nature of such communications. BAE Systems have developed a system ('stealthy wallpaper') which can block a single frequency range from being transmitted through a treated wall or ceiling to the neighbour. The problem posed to the Study Group was to investigate the maximum number of frequencies ensure the building is secure. The Study group found that this upper bound does not exist, so they were asked to find what are "good design-rules" so that an upper limit exists
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