TY - GEN

T1 - Sequences characterizing k-Trees

AU - Lotker, Zvi

AU - Majumdar, Debapriyo

AU - Narayanaswamy, N. S.

AU - Weber, Ingmar

PY - 2006/1/1

Y1 - 2006/1/1

N2 - A non-decreasing sequence of n integers is the degree sequence of a 1-tree (i.e., an ordinary tree) on n vertices if and only if there are least two 1's in the sequence, and the sum of the elements is 2(n - 1). We generalize this result in the following ways. First, a natural generalization of this statement is a necessary condition for k-trees, and we show that it is not sufficient for any k > 1. Second, we identify non-trivial sufficient conditions for the degree sequences of 2-trees. We also show that these sufficient conditions are almost necessary using bounds on the partition function p(n) and probabilistic methods. Third, we generalize the characterization of degrees of 1-trees in an elegant and counter-intuitive way to yield integer sequences that characterize k-trees, for all k.

AB - A non-decreasing sequence of n integers is the degree sequence of a 1-tree (i.e., an ordinary tree) on n vertices if and only if there are least two 1's in the sequence, and the sum of the elements is 2(n - 1). We generalize this result in the following ways. First, a natural generalization of this statement is a necessary condition for k-trees, and we show that it is not sufficient for any k > 1. Second, we identify non-trivial sufficient conditions for the degree sequences of 2-trees. We also show that these sufficient conditions are almost necessary using bounds on the partition function p(n) and probabilistic methods. Third, we generalize the characterization of degrees of 1-trees in an elegant and counter-intuitive way to yield integer sequences that characterize k-trees, for all k.

UR - http://www.scopus.com/inward/record.url?scp=33749561455&partnerID=8YFLogxK

U2 - 10.1007/11809678_24

DO - 10.1007/11809678_24

M3 - Conference contribution

AN - SCOPUS:33749561455

SN - 3540369252

SN - 9783540369257

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 216

EP - 225

BT - Computing and Combinatorics - 12th Annual International Conference, COCOON 2006, Proceedings

PB - Springer Verlag

T2 - 12th Annual International Conference on Computing and Combinatorics, COCOON 2006

Y2 - 15 August 2006 through 18 August 2006

ER -