59 research outputs found
Internal Partitions of Regular Graphs
An internal partition of an -vertex graph is a partition of
such that every vertex has at least as many neighbors in its own part as in the
other part. It has been conjectured that every -regular graph with
vertices has an internal partition. Here we prove this for . The case
is of particular interest and leads to interesting new open problems on
cubic graphs. We also provide new lower bounds on and find new families
of graphs with no internal partitions. Weighted versions of these problems are
considered as well
Leapfrog in Posets
We consider the following solitaire game whose rules are reminiscent of the
children's game of leapfrog. The player is handed an arbitrary ordering
of the elements of a finite poset . At each
round an element may "skip over" the element in front of it, i.e. swap
positions with it. For example, if , then it is allowed to
move from to the ordering
. The player is to carry out
such steps as long as such swaps are possible. When there are several
consecutive pairs of elements that satisfy this condition, the player can
choose which pair to swap next. Does the order of swaps matter for the final
ordering or is it uniquely determined by the initial ordering? The reader may
guess correctly that the latter proposition is correct. What may be more
surprising, perhaps, is that this question is not trivial. The proof works by
constructing an appropriate system of invariants
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