A bipartite graph with non-unimodal independent set sequence

0.06 MB·English

by  Arnab Bhattacharyya

A bipartite graph with non-unimodal independent set sequence
Arnab Bhattacharyya∗ Jeff Kahn†
January 10, 2013
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2 Abstract

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a We show that the independent set sequence of a bipartite graph need not be unimodal.
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1 Introduction

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For agraphG= (V,E) andaninteger t > 0, leti (G)denotethenumberofindependentsetsofsize
t
C
t in G. (Recall that an independentset is a set of vertices spanning no edges.) The independent set
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h sequence of G is the sequence i(G) = (i (G))α(G), where α(G) is the size of a largest independent
t t t=0
a set in G.
m
It was conjectured by Levit and Mandrescu [LM06] that for any bipartite graph G, i(G) is
[
unimodal; that is, that there is a k for which

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v i (G) 6 i (G) 6 6 i (G) > i (G) > > i (G).
0 1 k k+1 α(G)
2 ··· ···
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Evidence in favor of this was given by Levit and Mandrescu [LM06] and by Galvin (in [Gal12],
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1 which got us interested in the problem).
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1 In this note, we disprove the conjecture:
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3 Theorem 1.1. There are bipartite graphs G for which i(G) is not unimodal.
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v: See [Sta89] for a general survey of unimodality and the stronger notion of log-concavity.
i
X
r 2 Counterexample
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Given positive integers a and b > a, let G = G(a,b) = (V,E) with: V = V V V , where
1 2 3
∪ ∪
V ,V ,V are disjoint, V = b a and V = V = a; and E consists of a complete bipartite graph
1 2 3 1 2 3
| | − | | | |
between V and V and a perfect matching between V and V .
1 2 2 3
Lemma 2.1. For every t > 0, it(G) = (2t −1)(cid:0)at(cid:1)+(cid:0)bt(cid:1).
Proof. Each independent set in G is a subset of either V V or V V . Among independent sets
1 3 2 3
of size t, the number of the firsttype is (cid:0)b(cid:1), the numberof∪the second∪type is 2t(cid:0)a(cid:1), and the number
t t
that are of both types (that is, that are subsets of V3) is (cid:0)at(cid:1). (cid:3)
∗DIMACS & Rutgers University. Email: [email protected].
†Rutgers University. Email: [email protected]
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We now assert that i(G) is not unimodal if a is large and (say) b = alog 3 . In this case, the
expressions (cid:0)bt(cid:1) and 2t(cid:0)at(cid:1) are maximized at t1 = b/2 and t2 = 2a/3+O(1⌊) resp2ec⌋tively (the overlap
(cid:0)a(cid:1) is negligible), with each maximum on the order of 3a/√a. On the other hand, each expression
t
is o(3a/√a) if t is at least ω(√a) from the maximizing value. In particular, i (G) is much smaller
t
for t = (t +t )/2 than for t t ,t , and so, i(G) is not unimodal.
1 2 1 2
∈ { }
For a concrete example, we may take a = 100 and b = 159, for which explicit calculation gives
i (G) = 49984570869694708771111099844838813533288847750,
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i (G) = 44836126125886591149869334343833780227595935550,
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i (G) = 47256780307562808533825730975714923168070091770.
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3 Remarks
The construction above can be generalized to show that (for bipartite G) i(G) can have arbitrarily
many local maxima. Given (positive) integers k and a,a ,...,a , let G = G(a,a ,...,a ) =
1 k 1 k
(A B,E) be the bipartite graph where: A = k A and B = k B , with all A ’s and B ’s
∪ ∪i=0 i ∪j=1 j i j
disjoint; A = B = B = = B = a and A = a for i > 0; and E consists of a perfect
0 1 2 k i i
| | | | | | ··· | | | |
matching between A and B for each j > 0, together with a complete bipartite graph between
0 j
A and B for all (i,j) with j 6 i. Then for a,a ,...,a large with all the k + 1 expressions
i j 1 k
2a1+···+ai(1+2k−i)a roughly equal, an analysis similar to the one above shows that i(G) has k+1
local maxima.
Inclosing,letusmentiontheveryinteresting,stillunsettledconjectureofAlavi,Malde,Schwenk
and Erd˝os [AMSE87] that trees and forests have unimodal independent set sequences.
References
[AMSE87] Y. Alavi, P.J. Malde, A.J. Schwenk, and P. Erd˝os. The vertex independence sequence
of a graph is not constrained. Congressus Numerantium, 58:15–23, 1987. 2
[Gal12] D. Galvin. The independent set sequence of regular bipartite graphs. Discrete Mathe-
matics, 312:2881–2892, 2012. 1
[LM06] V.E. Levit and E. Mandrescu. Partial unimodality for independence polynomials of
Ko¨nig-Egerva´ry graphs. Congressus Numerantium, 179:109–119, 2006. 1
[Sta89] R.P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and ge-
ometry. Annals of the New York Academy of Sciences, 576:500–535, 1989. 1
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