TY - JOUR

T1 - The Roller-Coaster Conjecture Revisited

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

N1 - Publisher Copyright:
© 2017, Springer Japan KK.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer in J Comb Theory 8:91–98, 1970). If G is a well-covered graph with at least two vertices, and G- v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples in On some subclasses of well-covered graphs. Ph.D. Thesis, Vanderbilt University, 1975). The graph G is λ-quasi-regularizable if λ> 0 and λ· | S| ≤ | N(S) | for every independent set S of G. It is known that every well-covered graph without isolated vertices is 1-quasi-regularizable (Berge in Ann Discret Math 12:31–44, 1982). The independence polynomialI(G;x)=∑k=0αskxk is the generating function of independent sets in a graph G (Gutman and Harary in Util Math 24:97–106, 1983), where α is the independence number of G. The Roller-Coaster Conjecture (Michael and Traves in Graphs Comb 19:403–411, 2003), saying that for every permutation σ of the set {…,α} there exists a well-covered graph G with the independence number α such that the coefficients (sk) of I(G; x) satisfy (Formula Presented.) has been validated in Cutler and Pebody (J Comb Theory A 145:25–35, 2017). In this paper we show that independence polynomials of λ-quasi-regularizable graphs are partially unimodal. More precisely, the coefficients of an upper part of I(G; x) are in non-increasing order. Based on this finding, we prove that the unconstrained part of the independence sequence is: (Formula Presented.) for well-covered graphs, and (Formula Presented.) for 1-well-covered graphs, where α stands for the independence number, and n is the cardinality of the vertex set.

AB - A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer in J Comb Theory 8:91–98, 1970). If G is a well-covered graph with at least two vertices, and G- v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples in On some subclasses of well-covered graphs. Ph.D. Thesis, Vanderbilt University, 1975). The graph G is λ-quasi-regularizable if λ> 0 and λ· | S| ≤ | N(S) | for every independent set S of G. It is known that every well-covered graph without isolated vertices is 1-quasi-regularizable (Berge in Ann Discret Math 12:31–44, 1982). The independence polynomialI(G;x)=∑k=0αskxk is the generating function of independent sets in a graph G (Gutman and Harary in Util Math 24:97–106, 1983), where α is the independence number of G. The Roller-Coaster Conjecture (Michael and Traves in Graphs Comb 19:403–411, 2003), saying that for every permutation σ of the set {…,α} there exists a well-covered graph G with the independence number α such that the coefficients (sk) of I(G; x) satisfy (Formula Presented.) has been validated in Cutler and Pebody (J Comb Theory A 145:25–35, 2017). In this paper we show that independence polynomials of λ-quasi-regularizable graphs are partially unimodal. More precisely, the coefficients of an upper part of I(G; x) are in non-increasing order. Based on this finding, we prove that the unconstrained part of the independence sequence is: (Formula Presented.) for well-covered graphs, and (Formula Presented.) for 1-well-covered graphs, where α stands for the independence number, and n is the cardinality of the vertex set.

KW - 1-Well-covered graph

KW - Corona of graphs

KW - Independence polynomial

KW - Independent set

KW - Well-covered graph

UR - http://www.scopus.com/inward/record.url?scp=85027321098&partnerID=8YFLogxK

U2 - 10.1007/s00373-017-1834-z

DO - 10.1007/s00373-017-1834-z

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AN - SCOPUS:85027321098

SN - 0911-0119

VL - 33

SP - 1499

EP - 1508

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 6

ER -