132 research outputs found
Some hard families of parameterised counting problems
We consider parameterised subgraph-counting problems of the following form:
given a graph G, how many k-tuples of its vertices have a given property? A
number of such problems are known to be #W[1]-complete; here we substantially
generalise some of these existing results by proving hardness for two large
families of such problems. We demonstrate that it is #W[1]-hard to count the
number of k-vertex subgraphs having any property where the number of distinct
edge-densities of labelled subgraphs that satisfy the property is o(k^2). In
the special case that the property in question depends only on the number of
edges in the subgraph, we give a strengthening of this result which leads to
our second family of hard problems.Comment: A few more minor changes. This version to appear in the ACM
Transactions on Computation Theor
The multicolour size-Ramsey number of powers of paths
Given a positive integer s, a graph G is s-Ramsey for a graph H, denoted G→(H)s, if every s-colouring of the edges of G contains a monochromatic copy of H. The s-colour size-Ramsey number rˆs(H) of a graph H is defined to be rˆs(H)=min{|E(G)|:G→(H)s}. We prove that, for all positive integers k and s, we have rˆs(Pnk)=O(n), where Pnk is the kth power of the n-vertex path Pn
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