6 research outputs found
A proof of the Upper Matching Conjecture for large graphs
We prove that the `Upper Matching Conjecture' of Friedland, Krop, and
Markstr\"om and the analogous conjecture of Kahn for independent sets in
regular graphs hold for all large enough graphs as a function of the degree.
That is, for every and every large enough divisible by , a union of
copies of the complete -regular bipartite graph maximizes the
number of independent sets and matchings of size for each over all
-regular graphs on vertices. To prove this we utilize the cluster
expansion for the canonical ensemble of a statistical physics spin model, and
we give some further applications of this method to maximizing and minimizing
the number of independent sets and matchings of a given size in regular graphs
of a given minimum girth
Chomp on generalized Kneser graphs and others
In chomp on graphs, two players alternatingly pick an edge or a vertex from a
graph. The player that cannot move any more loses. The questions one wants to
answer for a given graph are: Which player has a winning strategy? Can a
explicit strategy be devised? We answer these questions (and determine the
Nim-value) for the class of generalized Kneser graphs and for several families
of Johnson graphs. We also generalize some of these results to the clique
complexes of these graphs. Furthermore, we determine which player has a winning
strategy for some classes of threshold graphs.Comment: 17 pages, 4 figures, removed a wrong theorem about almost bipartite
graphs from a previous versio
Mechanising an algebraic rely-guarantee refinement calculus
PhD ThesisDespite rely-guarantee (RG) being a well-studied program logic established in the 1980s, it
was not until recently that researchers realised that rely and guarantee conditions could be
treated as independent programming constructs. This recent reformulation of RG paved the
way to algebraic characterisations which have helped to better understand the difficulties that
arise in the practical application of this development approach.
The primary focus of this thesis is to provide automated tool support for a rely-guarantee
refinement calculus proposed by Hayes et. al., where rely and guarantee are defined as
independent commands. Our motivation is to investigate the application of an algebraic
approach to derive concrete examples using this calculus. In the course of this thesis, we
locate and fix a few issues involving the refinement language, its operational semantics and
preexisting proofs. Moreover, we extend the refinement calculus of Hayes et. al. to cover
indexed parallel composition, non-atomic evaluation of expressions within specifications,
and assignment to indexed arrays. These extensions are illustrated via concrete examples.
Special attention is given to design decisions that simplify the application of the mechanised
theory. For example, we leave part of the design of the expression language on the
hands of the user, at the cost of the requiring the user to define the notion of undefinedness
for unary and binary operators; and we also formalise a notion of indexed parallelism that is
parametric on the type of the indexes, this is done deliberately to simplify the formalisation of
algorithms. Additionally, we use stratification to reduce the number of cases in in simulation
proofs involving the operational semantics. Finally, we also use the algebra to discuss the
role of types in program derivation
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum