22 research outputs found
Friends and Strangers Walking on Graphs
Given graphs and with vertex sets and of the same
cardinality, we define a graph whose vertex set consists of
all bijections , where two bijections and
are adjacent if they agree everywhere except for two adjacent
vertices such that and are adjacent in
. This setup, which has a natural interpretation in terms of friends and
strangers walking on graphs, provides a common generalization of Cayley graphs
of symmetric groups generated by transpositions, the famous -puzzle,
generalizations of the -puzzle as studied by Wilson, and work of Stanley
related to flag -vectors. We derive several general results about the graphs
before focusing our attention on some specific choices of
. When is a path graph, we show that the connected components of
correspond to the acyclic orientations of the complement of
. When is a cycle, we obtain a full description of the connected
components of in terms of toric acyclic orientations of the
complement of . We then derive various necessary and/or sufficient
conditions on the graphs and that guarantee the connectedness of
. Finally, we raise several promising further questions.Comment: 28 pages, 5 figure
Cats in cubes
Answering a recent question of Patchell and Spiro, we show that when a
-dimensional cube of side length is filled with letters, the word
can appear contiguously at most times (allowing
diagonals); we also characterize when equality occurs and extend our results to
words other than