A Study of Graphical Permutations
Jessica Thune*
Department of Mathematical Sciences, University of Nevada Las Vegas
Las Vegas, NV 89154
thunej@unlv.nevada.edu
A permutation π on a set of positive integers {a1 , a2 , ..., an } is said to be graphical if there exists
a graph containing exactly ai vertices of degree π(ai ). It has been shown that for positive integers
with a1 < a2 < ... < an , if π(an ) = an , then the permutation π is graphical if and only if the sum
n−1
n
P
P
ai π(ai ). This known result has been proved using a criterion of
ai π(ai ) is even and an ≤
i=1
i=1
Fulkerson, Hoffman, and McAndrew which requires the verification of 12 n(n+1) inequalities. In this
talk, we use a criterion of Tripathi and Vijay to give a shorter proof of this result, requiring only the
verification of n inequalities. We also use this criterion to provide a similar result for permutations
π such that π(an−1 ) = an . We prove that such a permutation is graphical if and only if the sum
n
P
P
ai π(ai ) is even and an an−1 ≤ an−1 (an−1 − 1) +
ai π(ai ). We also consider permutations
i=1
i6=n−1
such that π(an ) = an−1 , and then more generally, permutations such that π(an ) = an−j for some
j < n.