60 research outputs found
An Exact Algorithm for the Generalized List -Coloring Problem
The generalized list -coloring is a common generalization of many graph
coloring models, including classical coloring, -labeling, channel
assignment and -coloring. Every vertex from the input graph has a list of
permitted labels. Moreover, every edge has a set of forbidden differences. We
ask for such a labeling of vertices of the input graph with natural numbers, in
which every vertex gets a label from its list of permitted labels and the
difference of labels of the endpoints of each edge does not belong to the set
of forbidden differences of this edge. In this paper we present an exact
algorithm solving this problem, running in time ,
where is the maximum forbidden difference over all edges of the input
graph and is the number of its vertices. Moreover, we show how to improve
this bound if the input graph has some special structure, e.g. a bounded
maximum degree, no big induced stars or a perfect matching
An exact algorithm for graph coloring with polynomial memory
In this paper, we give an algorithm that computes the chromatic number of a
graph in O(5.283n) time and polynomial memory
On the Number of Maximal Bipartite Subgraphs of a Graph
We show new lower and upper bounds on the number of maximal induced bipartite subgraphs of graphs with n vertices. We present an infinite family of graphs having 105^{n/10} ~= 1.5926^n such subgraphs, which improves an earlier lower bound by Schiermeyer (1996). We show an upper bound of n . 12^{n/4} ~= n . 1.8613^n and give an algorithm that lists all maximal induced bipartite subgraphs in time proportional to this bound. This is used in an algorithm for checking 4-colourability of a graph running within the same time bound
Coloring, location and domination of corona graphs
A vertex coloring of a graph is an assignment of colors to the vertices
of such that every two adjacent vertices of have different colors. A
coloring related property of a graphs is also an assignment of colors or labels
to the vertices of a graph, in which the process of labeling is done according
to an extra condition. A set of vertices of a graph is a dominating set
in if every vertex outside of is adjacent to at least one vertex
belonging to . A domination parameter of is related to those structures
of a graph satisfying some domination property together with other conditions
on the vertices of . In this article we study several mathematical
properties related to coloring, domination and location of corona graphs.
We investigate the distance- colorings of corona graphs. Particularly, we
obtain tight bounds for the distance-2 chromatic number and distance-3
chromatic number of corona graphs, throughout some relationships between the
distance- chromatic number of corona graphs and the distance- chromatic
number of its factors. Moreover, we give the exact value of the distance-
chromatic number of the corona of a path and an arbitrary graph. On the other
hand, we obtain bounds for the Roman dominating number and the
locating-domination number of corona graphs. We give closed formulaes for the
-domination number, the distance- domination number, the independence
domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page
Lower Bounds for the Graph Homomorphism Problem
The graph homomorphism problem (HOM) asks whether the vertices of a given
-vertex graph can be mapped to the vertices of a given -vertex graph
such that each edge of is mapped to an edge of . The problem
generalizes the graph coloring problem and at the same time can be viewed as a
special case of the -CSP problem. In this paper, we prove several lower
bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main
result is a lower bound .
This rules out the existence of a single-exponential algorithm and shows that
the trivial upper bound is almost asymptotically
tight.
We also investigate what properties of graphs and make it difficult
to solve HOM. An easy observation is that an upper
bound can be improved to where
is the minimum size of a vertex cover of . The second
lower bound shows that the upper bound is
asymptotically tight. As to the properties of the "right-hand side" graph ,
it is known that HOM can be solved in time and
where is the maximum degree of
and is the treewidth of . This gives
single-exponential algorithms for graphs of bounded maximum degree or bounded
treewidth. Since the chromatic number does not exceed
and , it is natural to ask whether similar
upper bounds with respect to can be obtained. We provide a negative
answer to this question by establishing a lower bound for any
function . We also observe that similar lower bounds can be obtained for
locally injective homomorphisms.Comment: 19 page
Faster Graph Coloring in Polynomial Space
We present a polynomial-space algorithm that computes the number independent
sets of any input graph in time for graphs with maximum degree 3
and in time for general graphs, where n is the number of
vertices. Together with the inclusion-exclusion approach of Bj\"orklund,
Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster
polynomial-space algorithm for the graph coloring problem with running time
. As a byproduct, we also obtain an exponential-space
time algorithm for counting independent sets. Our main algorithm
counts independent sets in graphs with maximum degree 3 and no vertex with
three neighbors of degree 3. This polynomial-space algorithm is analyzed using
the recently introduced Separate, Measure and Conquer approach [Gaspers &
Sorkin, ICALP 2015]. Using Wahlstr\"om's compound measure approach, this
improvement in running time for small degree graphs is then bootstrapped to
larger degrees, giving the improvement for general graphs. Combining both
approaches leads to some inflexibility in choosing vertices to branch on for
the small-degree cases, which we counter by structural graph properties
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