469 research outputs found
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
Injective colorings of sparse graphs
Let denote the maximum average degree (over all subgraphs) of
and let denote the injective chromatic number of . We prove that
if , then ; and if , then . Suppose that is a planar graph with
girth and . We prove that if , then
; similarly, if , then
.Comment: 10 page
Measurable versions of Vizing's theorem
We establish two versions of Vizing's theorem for Borel multi-graphs whose
vertex degrees and edge multiplicities are uniformly bounded by respectively
and . The ``approximate'' version states that, for any Borel
probability measure on the edge set and any , we can properly
colour all but -fraction of edges with colours in a
Borel way. The ``measurable'' version, which is our main result, states that
if, additionally, the measure is invariant, then there is a measurable proper
edge colouring of the whole edge set with at most colours
A Proof of a Conjecture of Ohba
We prove a conjecture of Ohba which says that every graph on at most
vertices satisfies .Comment: 21 page
A general framework for coloring problems: old results, new results, and open problems
In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants
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