98 research outputs found
Full Orientability of the Square of a Cycle
Let D be an acyclic orientation of a simple graph G. An arc of D is called
dependent if its reversal creates a directed cycle. Let d(D) denote the number
of dependent arcs in D. Define m and M to be the minimum and the maximum number
of d(D) over all acyclic orientations D of G. We call G fully orientable if G
has an acyclic orientation with exactly k dependent arcs for every k satisfying
m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n
is fully orientable except n=6.Comment: 7 pages, accepted by Ars Combinatoria on May 26, 201
Chordal Graphs are Fully Orientable
Suppose that D is an acyclic orientation of a graph G. An arc of D is called
dependent if its reversal creates a directed cycle. Let m and M denote the
minimum and the maximum of the number of dependent arcs over all acyclic
orientations of G. We call G fully orientable if G has an acyclic orientation
with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is
called chordal if every cycle in G of length at least four has a chord. We show
that all chordal graphs are fully orientable.Comment: 11 pages, 1 figure, accepted by Ars Combinatoria (March 26, 2010
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The frequency assignment problem
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight.
We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EPSR
Complete Acyclic Colorings
We study two parameters that arise from the dichromatic number and the
vertex-arboricity in the same way that the achromatic number comes from the
chromatic number. The adichromatic number of a digraph is the largest number of
colors its vertices can be colored with such that every color induces an
acyclic subdigraph but merging any two colors yields a monochromatic directed
cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest
number of colors that can be used such that every color induces a forest but
merging any two yields a monochromatic cycle. We study the relation between
these parameters and their behavior with respect to other classical parameters
such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure
Colouring Complete Multipartite and Kneser-type Digraphs
The dichromatic number of a digraph is the smallest such that can
be partitioned into acyclic subdigraphs, and the dichromatic number of an
undirected graph is the maximum dichromatic number over all its orientations.
Extending a well-known result of Lov\'{a}sz, we show that the dichromatic
number of the Kneser graph is and that the
dichromatic number of the Borsuk graph is if is large
enough. We then study the list version of the dichromatic number. We show that,
for any and , the list
dichromatic number of is . This extends a recent
result of Bulankina and Kupavskii on the list chromatic number of ,
where the same behaviour was observed. We also show that for any ,
and , the list dichromatic number of the complete
-partite graph with vertices in each part is , extending
a classical result of Alon. Finally, we give a directed analogue of Sabidussi's
theorem on the chromatic number of graph products.Comment: 15 page
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