10 research outputs found
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Improper choosability and Property B
A fundamental connection between list vertex colourings of graphs and
Property B (also known as hypergraph 2-colourability) was already known to
Erd\H{o}s, Rubin and Taylor. In this article, we draw similar connections for
improper list colourings. This extends results of Kostochka, Alon, and Kr\'al'
and Sgall for, respectively, multipartite graphs, graphs of large minimum
degree, and list assignments with bounded list union.Comment: 12 page
On the choosability of -minor-free graphs
Given a graph , let us denote by and ,
respectively, the maximum chromatic number and the maximum list chromatic
number of -minor-free graphs. Hadwiger's famous coloring conjecture from
1943 states that for every . In contrast, for list
coloring it is known that
and thus, is bounded away from the conjectured value for
by at least a constant factor. The so-called -Hadwiger's
conjecture, proposed by Seymour, asks to prove that
for a given graph (which would be implied by Hadwiger's conjecture). In
this paper, we prove several new lower bounds on , thus exploring
the limits of a list coloring extension of -Hadwiger's conjecture. Our main
results are:
For every and all sufficiently large graphs we have
, where
denotes the vertex-connectivity of .
For every there exists such that
asymptotically almost every -vertex graph with edges satisfies .
The first result generalizes recent results on complete and complete
bipartite graphs and shows that the list chromatic number of -minor-free
graphs is separated from the natural lower bound by a
constant factor for all large graphs of linear connectivity. The second
result tells us that even when is a very sparse graph (with an average
degree just logarithmic in its order), can still be separated from
by a constant factor arbitrarily close to . Conceptually
these results indicate that the graphs for which is close to
are typically rather sparse.Comment: 14 page
Defective Coloring on Classes of Perfect Graphs
In Defective Coloring we are given a graph and two integers ,
and are asked if we can -color so that the maximum
degree induced by any color class is at most . We show that this
natural generalization of Coloring is much harder on several basic graph
classes. In particular, we show that it is NP-hard on split graphs, even when
one of the two parameters , is set to the smallest possible
fixed value that does not trivialize the problem ( or ). Together with a simple treewidth-based DP algorithm this completely
determines the complexity of the problem also on chordal graphs. We then
consider the case of cographs and show that, somewhat surprisingly, Defective
Coloring turns out to be one of the few natural problems which are NP-hard on
this class. We complement this negative result by showing that Defective
Coloring is in P for cographs if either or is fixed; that
it is in P for trivially perfect graphs; and that it admits a sub-exponential
time algorithm for cographs when both and are unbounded
Improper colourings inspired by Hadwiger’s conjecture
Hadwiger’s Conjecture asserts that every Kt-minor-free graph has a proper (t − 1)-colouring. We relax the conclusion in Hadwiger’s Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t − 2)-colourable with monochromatic components of order at most 1/2 (t − 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t − 1)-colourable with monochromatic degree at most t − 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree
Parameterized Algorithms for Graph Partitioning Problems
In parameterized complexity, a problem instance (I, k) consists of an input I and an
extra parameter k. The parameter k usually a positive integer indicating the size of the
solution or the structure of the input. A computational problem is called fixed-parameter
tractable (FPT) if there is an algorithm for the problem with time complexity O(f(k).nc
),
where f(k) is a function dependent only on the input parameter k, n is the size of the
input and c is a constant. The existence of such an algorithm means that the problem
is tractable for fixed values of the parameter. In this thesis, we provide parameterized
algorithms for the following NP-hard graph partitioning problems:
(i) Matching Cut Problem: In an undirected graph, a matching cut is a partition
of vertices into two non-empty sets such that the edges across the sets induce a matching.
The matching cut problem is the problem of deciding whether a given graph has
a matching cut. The Matching Cut problem is expressible in monadic second-order
logic (MSOL). The MSOL formulation, together with Courcelle’s theorem implies linear
time solvability on graphs with bounded tree-width. However, this approach leads to a
running time of f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the
tree-width of the graph and n is the number of vertices of the graph. The dependency of
f(||ϕ||, t) on ||ϕ|| can be as bad as a tower of exponentials.
In this thesis we give a single exponential algorithm for the Matching Cut problem
with tree-width alone as the parameter. The running time of the algorithm is 2O(t)
· n.
This answers an open question posed by Kratsch and Le [Theoretical Computer Science,
2016]. We also show the fixed parameter tractability of the Matching Cut problem
when parameterized by neighborhood diversity or other structural parameters.
(ii) H-Free Coloring Problems: In an undirected graph G for a fixed graph H,
the H-Free q-Coloring problem asks to color the vertices of the graph G using at
most q colors such that none of the color classes contain H as an induced subgraph.
That is every color class is H-free. This is a generalization of the classical q-Coloring
problem, which is to color the vertices of the graph using at most q colors such that no
pair of adjacent vertices are of the same color. The H-Free Chromatic Number is
the minimum number of colors required to H-free color the graph.
For a fixed q, the H-Free q-Coloring problem is expressible in monadic secondorder
logic (MSOL). The MSOL formulation leads to an algorithm with time complexity
f(||ϕ||, t) · n, where ||ϕ|| is the length of the MSOL formula, t is the tree-width of the
graph and n is the number of vertices of the graph.
In this thesis we present the following explicit combinatorial algorithms for H-Free
Coloring problems:
• An O(q
O(t
r
)
· n) time algorithm for the general H-Free q-Coloring problem,
where r = |V (H)|.
• An O(2t+r log t
· n) time algorithm for Kr-Free 2-Coloring problem, where Kr is
a complete graph on r vertices.
The above implies an O(t
O(t
r
)
· n log t) time algorithm to compute the H-Free Chromatic
Number for graphs with tree-width at most t. Therefore H-Free Chromatic
Number is FPT with respect to tree-width.
We also address a variant of H-Free q-Coloring problem which we call H-(Subgraph)Free
q-Coloring problem, which is to color the vertices of the graph such that none of the
color classes contain H as a subgraph (need not be induced).
We present the following algorithms for H-(Subgraph)Free q-Coloring problems.
• An O(q
O(t
r
)
· n) time algorithm for the general H-(Subgraph)Free q-Coloring
problem, which leads to an O(t
O(t
r
)
· n log t) time algorithm to compute the H-
(Subgraph)Free Chromatic Number for graphs with tree-width at most t.
• An O(2O(t
2
)
· n) time algorithm for C4-(Subgraph)Free 2-Coloring, where C4
is a cycle on 4 vertices.
• An O(2O(t
r−2
)
· n) time algorithm for {Kr\e}-(Subgraph)Free 2-Coloring,
where Kr\e is a graph obtained by removing an edge from Kr.
• An O(2O((tr2
)
r−2
)
· n) time algorithm for Cr-(Subgraph)Free 2-Coloring problem,
where Cr is a cycle of length r.
(iii) Happy Coloring Problems: In a vertex-colored graph, an edge is happy if its
endpoints have the same color. Similarly, a vertex is happy if all its incident edges are
happy. we consider the algorithmic aspects of the following Maximum Happy Edges
(k-MHE) problem: given a partially k-colored graph G, find an extended full k-coloring
of G such that the number of happy edges are maximized. When we want to maximize
the number of happy vertices, the problem is known as Maximum Happy Vertices
(k-MHV).
We show that both k-MHE and k-MHV admit polynomial-time algorithms for trees.
We show that k-MHE admits a kernel of size k + `, where ` is the natural parameter,
the number of happy edges. We show the hardness of k-MHE and k-MHV for some
special graphs such as split graphs and bipartite graphs. We show that both k-MHE
and k-MHV are tractable for graphs with bounded tree-width and graphs with bounded
neighborhood diversity.
vii
In the last part of the thesis we present an algorithm for the Replacement Paths
Problem which is defined as follows: Let G (|V (G)| = n and |E(G)| = m) be an undirected
graph with positive edge weights. Let PG(s, t) be a shortest s − t path in G. Let l be the
number of edges in PG(s, t). The Edge Replacement Path problem is to compute a
shortest s − t path in G\{e}, for every edge e in PG(s, t). The Node Replacement
Path problem is to compute a shortest s−t path in G\{v}, for every vertex v in PG(s, t).
We present an O(TSP T (G) + m + l
2
) time and O(m + l
2
) space algorithm for both
the problems, where TSP T (G) is the asymptotic time to compute a single source shortest
path tree in G. The proposed algorithm is simple and easy to implement
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..