TY - JOUR

T1 - Enumeration of the degree sequences of non-separable graphs and connected graphs

AU - Rødseth, Øystein J.

AU - Sellers, James A.

AU - Tverberg, Helge

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2009/7

Y1 - 2009/7

N2 - In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d1, d2, ..., dn to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions dn s (2 m) and dc (2 m), the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: dn s (2 m) = p (2 m) - p (2 m - 1) - underover(∑, j = 0, m - 2) p (j) and dc (2 m) = p (2 m) - p (m - 1) - 2 underover(∑, j = 0, m - 2) p (j) where p (j) is the number of unrestricted integer partitions of j.

AB - In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d1, d2, ..., dn to be the degree sequence of a non-separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions dn s (2 m) and dc (2 m), the number of degree sequences with degree sum 2 m representable by non-separable graphs and connected graphs (respectively). Indeed, we give both generating function proofs as well as bijective proofs of the following identities: dn s (2 m) = p (2 m) - p (2 m - 1) - underover(∑, j = 0, m - 2) p (j) and dc (2 m) = p (2 m) - p (m - 1) - 2 underover(∑, j = 0, m - 2) p (j) where p (j) is the number of unrestricted integer partitions of j.

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U2 - 10.1016/j.ejc.2008.10.006

DO - 10.1016/j.ejc.2008.10.006

M3 - Article

AN - SCOPUS:63149117870

VL - 30

SP - 1309

EP - 1317

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

IS - 5

ER -