3,012 research outputs found
Questions on the Structure of Perfect Matchings inspired by Quantum Physics
We state a number of related questions on the structure of perfect matchings.
Those questions are inspired by and directly connected to Quantum Physics. In
particular, they concern the constructability of general quantum states using
modern photonic technology. For that we introduce a new concept, denoted as
inherited vertex coloring. It is a vertex coloring for every perfect matching.
The colors are inherited from the color of the incident edge for each perfect
matching. First, we formulate the concepts and questions in pure
graph-theoretical language, and finally we explain the physical context of
every mathematical object that we use. Importantly, every progress towards
answering these questions can directly be translated into new understanding in
quantum physics.Comment: 10 pages, 4 figures, 6 questions (added suggestions from peer-review
Recoloring bounded treewidth graphs
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Any graph is
-mixing, where is the treewidth of the graph (Cereceda 2006). We
prove that the shortest sequence between any two -colorings is at most
quadratic, a problem left open in Bonamy et al. (2012).
Jerrum proved that any graph is -mixing if is at least the maximum
degree plus two. We improve Jerrum's bound using the grundy number, which is
the worst number of colors in a greedy coloring.Comment: 11 pages, 5 figure
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