a re–ned version of the roller-coaster conjecture mandrescu.pdf · independent sets and...

A refined version of the Roller-Coaster Conjecture

Vadim E. LevitAriel University, Israel

&Eugen Mandrescu

Holon Institute of Technology, Israel

May 24-28, 2017Ninth Shanghai Conference on Combinatorics (9SHCC)

Shanghai Jiaotong University, Shanghai, China

Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 1 / 50

Outline

Independent sets and independence polynomials

The Chaotic Theorem (due to Alavi, Malde, Shwenk and Erdös)

The Unimodalty Conjecture for Trees (Alavi et al.)

Well-covered graphs and some corresponding conjectures

Very well-covered graphs and some corresponding conjectures

1-well-covered graphs and some corresponding conjectures

λ-quasi-regularizable graphs and their connections with the subject

Some open problems


Independent sets and independence number

DefinitionAn independent set is a set of pairwise non-adjacent vertices.

The independence number of G is the size α(G ) of

a largest (i.e., maximum) independent set in G .

Example

{a}, {a, b}, {a, b, x}, {a, b, c , y} are independent sets of G{u, v} is a maximal independent set{a, b, c , y} is a maximum independent set , hence α(G ) = 4

w w@@@w

a

b

c

x

u v y

ww w wG w wFigure: α(G ) = 4 = |{a, b, c , y}|Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 3 / 50

Independence polynomial of a graph

Definition (I. Gutman & F. Harary, Utilitas Mathematica 1983)If G has sk independent sets of size k, then

I (G ; x) = s0 + s1x + s2x2 + ...+ sαxα, α = α (G ) ,

is called the independence polynomial of G .

Examples:

w w ww

G w w ww@@@

H

Figure: I (G ; x) = 1+ 4x + 3x2 + x3, while I (H; x) = 1+ 4x + 2x2.

The disjoint union G ∪H of the graphs G ,H has

I (G ∪H; x) =(1+ 4x + 3x2 + x3

)•(1+ 4x + 2x2

)Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 4 / 50

Zykov sum of graphs and its independence polynomial

Definition (A. A. Zykov, 1990)The Zykov sum of two disjoint graphs G and H is the graph G +H

obtained by joining each vertex of G with each vertex of H by an edge.

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HHHHHH

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wwK2wwwP3

G1

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HHHHHH

��

��HHHHHH

@@@@@@

G2

wwwP3

wwwK3

Figure: G1 = K2 + P3 and G2 = P3 +K3.

Fact (Gutman and Harary, ’83): I (G +H; x) = I (G ; x) + I (H; x)− 1

I (P3 +K3; x) =(1+ 3x + x2

)+ (1+ 3x)− 1 = 1+ 6x + x2.


Corona of graphs and its independence polynomial

Definition (R. Frucht and F. Harary, Aequationes Math. 1970)

The corona G ◦H of G and H is obtained from G and |V (G )| copiesof H, such that each vertex of G is joined to all vertices of a copy of H.

Theorem (I. Gutman, 1992)

I (G ◦H; x) = (I (H; x))n • I(G ; x

I (H ;x )

), with n = |V (G )|.

Example

I (P3 ◦K2; x) = (I (K2; x))3 • I(P3; x

I (K2;x )

)=

= (1+ 2x)3 • I(P3; x

1+2x

)= 1+ 9x + 25x2 + 22x3

w w wa b c

P3

w wK2 �

��

�� @

@@

w w wa b cww w w w wP3 ◦K2P3 ◦K2P3 ◦K2

Figure: G = P3,H = K2 and G ◦H = P3 ◦K2.Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 6 / 50

DefinitionGiven a graph G , its line graph L(G ) is the graph whose vertex set isthe edge set of G , and two vertices are adjacent if they share an end in G .

Matching polynomial M(G ; x) of G ≡ Independence poly of L(G )Unlike M(G ; x), the polynomial I (G ; x) may have non-real roots.L(G ) is a claw-free graph (i.e., a K1,3-free graph)!

Theorem (M. Chudnovsky and P. Seymour, J. Comb Th 2007)

If G is a claw-free graph, then I (G ; x) has only real roots.

Example

M(G ; x) = I (L(G ); x) = 1+ 6x + 7x2 + x3.

c b

d

a

few w w ww w

G��

��

��@@@w w w w

w wf

b

c da

e

f

b

c da

eL(G )L(G )

Figure:Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 7 / 50

Some general facts about polynomials

Let P (x) = a0 + a1x + ...+ anxn be a polynomial with all ak > 0

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?

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P has all its roots realP has all its roots real

Newton’s inequality: a2k ≥ ak−1 • ak+1 • k+1k •n−k+1n−k for 1 ≤ k ≤ n− 1Newton’s inequality: a2k ≥ ak−1 • ak+1 • k+1k •n−k+1n−k for 1 ≤ k ≤ n− 1

P is log-concave, i.e., a2k ≥ ak−1 • ak+1 for 1 ≤ k ≤ n− 1P is log-concave, i.e., a2k ≥ ak−1 • ak+1 for 1 ≤ k ≤ n− 1

I. Newton, Arithmetica universalis, 1707

folklore

P is unimodal, i.e., a0 ≤ ... ≤ aj ≥ ... ≥ an for some j (= mode)P is unimodal, i.e., a0 ≤ ... ≤ aj ≥ ... ≥ an for some j (= mode)


Examples

I (K10 + 6K1; x) = (1+ x)6 + 10x =

= 1+ 16x + 15x2 + 20x3 + 15x4 + 6x5 + x6 is not unimodal

I (K43 + 3K7; x) = 1+ 64x + 147x2 + 343x3 is not log-concave

I (K1,3; x) = 1+ 4x + 3x2 + x3 is log-concave;

hence it is unimodal, but has non-real roots

I (C7; x) = 1+ 7x + 14x2 + 7x3 = 0 has only real roots,

hence it is log-concave and unimodal as well

K10

www

wwwK10 + 6K1K10 + 6K1

wwww

��

K1,3

w w w ww w w

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C7


If I (G ; x) = s0 + s1x + s2x2 + ...+ sαxα, where α = α (G ),

then (s1, s2, ..., sα) is the independence sequence of G .

Theorem (Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös, 1987)

For every permutation π of {1, 2, ..., α}, there is a graph Gwith α(G ) = α, such that sπ(1) < sπ(2) < ... < sπ(α),

i.e., for general graphs, the independence sequence is unconstrained.

Conjecture: I (T ; x) is unimodal for any tree T (Alavi et al., 1987).

Example (K. Dohmen, A. Ponitz and P. Tittmann, DMTCS 2003))

I (T1; x) = 1+ 10x + 36x2 + 58x3 + 42x4 + 12x5 + x6 = I (T2; x)

v v v v v vv v v v

T1 v v v v v vv v vv

T2


Is I (G ; x) unimodal, i.e., is there some k ∈ {0, 1, ..., n},such that s0 ≤ ... ≤ sk−1 ≤ sk ≥ sk+1 ≥ ... ≥ sα(G )?

Theorem (Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös, 1987)For general graphs, the independence sequence is unconstrained.


Some "Pro’s": (Levit and Mandrescu, 2007)

(i) sd(2α−1)/3e ≥ ... ≥ sα−1 ≥ sα hold for any tree T with α(T ) = α;

(ii) If T = H ◦K1 is a tree and α(T ) = α, then

(ii.1) s0 ≤ s1 ≤ ... ≤ sdα/2e and sd(2α−1)/3e ≥ ..... ≥ sα−1 ≥ sα,

(ii.2) I (T ; x) is unimodal, for α ≤ 9.


Well-Covered Graphs

Definition (M. D. Plummer, J Comb Th 1970)G is well-covered if all its maximal independent sets have the same size.

The only well-covered cycles are C3,C4,C5 and C7.

girth = the length of a shortest cycle, and = ∞ if no cycles.

Theorem (A. Finbow, B. Hartnell, R. Nowakowski, J Comb Th 1993)

A connected graph G of girth ≥ 6, and C7 6= G 6= K1, is well-covered iffits pendant edges form a perfect matching, i.e., G = H ◦K1 for some H.

w w ww

��

G1 wv w w ww w

@@@

G2G2

w w wG3G3 w w w

Figure: Only G1 is NOT well-covered. G3 = P3 ◦K1


Conjecture (J. I. Brown, K. Dilcher, R. J. Nowakowski, Journal ofAlgebraic Combinatorics 2000)

If G is well-covered, then I (G ; x) is unimodal.

TRUE for α(G ) ≤ 3 and FALSE for 4 ≤ α(G ) ≤ 7

(T. S. Michael and W. N. Traves, Graphs & Combinatorics 2003)

FALSE for α(G ) ≥ 4 (Levit & Mandrescu, European J Comb 2006)

TRUE for very well-covered graphs with α(G ) ≤ 9(Levit and Mandrescu, in Graph Theory in Paris 2006)

Recall that: a well covered graph G of order 2α (G ) and

without isolated vertices is called very well-covered

(O. Favaron, DM 1982).


Fact (T. S. Michael, W. N. Traves, Graphs & Comb 2003)There are well-cov graphs with non-unimodal independence polynomials.

Examples (Levit and Mandrescu, European J. Comb 2006)

Gq = (qK1000·q) ∪ ((4K10) +K1701(4)) and Gq + Gq are well-coveredgraphs (with α = q + 4), and their indep polynomials are not unimodal.

q times︷︸︸︷&%'$K1000·q w w w w &%

'$K1000·q

��

��

K10 K10

'&

$%

K1701(4)

��

K10 K10GqGqGq

Figure: Well-covered graphs with non-unimodal independence polynomials.


Theorem (T. S. Michael and W. N. Traves, Graphs & Comb 2003;Levit & Mandrescu, DAM 2008)

For well-covered graphs, the independence sequence (s1, s2, · · · , sα(G ))satisfies s0 ≤ s1 ≤ · · · ≤ sdα(G )/2e.

Roller-Coaster Conjecture:

Conjecture (T. S. Michael, W. N. Traves, Graphs & Comb 2003)

For every permutation π of the set {dα/2e , dα/2e+ 1, ..., α},

there exists some well-covered graph G, with α(G ) = α, such that

sπ(dα/2e) < sπ(dα/2e+1) < · · · < sπ(α).

TRUE for α(G ) ≤ 7 (Michael & Traves, Graphs & Comb 2003)TRUE for α(G ) ≤ 11 (P. Matchett, The Electronic J Comb 2004)VALIDATED (J. Cutler and L. Pebody, J. of Comb. Th. A 2017)


Theorem (T. S. Michael and W. N. Traves, Graphs & Comb 2003;Levit & Mandrescu, DAM 2008)

For well-covered graphs, the independence sequence (s1, s2, · · · , sα(G ))satisfies s0 ≤ s1 ≤ · · · ≤ sdα(G )/2e.

Conjecture (Roller-Coaster, Michael and Traves, 2003)

For well-covered graphs, the part

(sdα(G )/2e, sdα(G )/2e+1, · · · , sα(G )) of the independence sequence

is unconstrained in the sense of Alavi et al.

TRUE for α(G ) ≤ 7 (Michael & Traves, Graphs & Comb 2003)TRUE for α(G ) ≤ 11 (P. Matchett, The Electronic J Comb 2004)VALIDATED (J. Cutler and L. Pebody, J. of Comb. Th. A 2017)


TheoremIf G is a well-covered graph without isolated vertices, then

s0 ≤ s1 ≤ · · · ≤ sdα/2e and [NEW:] sr ≥ sr+1 ≥ · · · ≥ sα,for r = min

{α,⌈ n−1

3

⌉}, where |V (G )| = n and α(G ) = α.

Actually, min{

α,⌈ n−1

3

⌉}< α only for n ≤ 3α− 2.

ExampleG is well-covered with α = 4, n = 9 < 3α− 2 = 10 and satisfies:

s0 = 1 ≤ s1 = 9 ≤ s2 = 27 = sdα/2e and sr = s3 = 32 ≥ 13 = s4 = sα

as r = min{

α,⌈ n−1

3

⌉}= 3 and I (G ) = 1+ 9x + 27x2 + 32x3 + 13x4.

w w w ww w w w w

��

G

Figure: G is well-covered with α (G ) = 4 and |V (G )| = n = 9Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 17 / 50

Our Refinement



{α,⌈ n−1

3

⌉}, where |V (G )| = n and α(G ) = α.

Actually, min{

α,⌈ n−1

3


ConjectureLet α ≥ 2, n ≥ 4, and 2α ≤ n ≤ 3α− 2. Then, for every permutation σ

of the set {⌈

α2

⌉,⌈

α2

⌉+ 1, ...,

⌈ n−13

⌉}, there exists a well-covered graph

G without isolated vertices, having α(G ) = α and |V (G )| = n,such that its independence sequence satisfies

sσ(d α2 e) < sσ(d α

2 e+1) < · · · < sσ(d n−13 e).


Our Refinement



{α,⌈ n−1

3

⌉}, where |V (G )| = n and α(G ) = α.

Actually, min{

α,⌈ n−1

3


Conjecture

For well-covered graphs on n vertices, where n ≤ 3α− 2,

the part (sd α2 e, sd α

2 e+1, ..., sd n−13 e) of the independence sequence

is unconstrained in the sense of Alavi et al.


The well-covered graphs used to validate the Roller-CoasterConjecture are of rather large order, and even Cutler and Pebodythink that it is “far from the minimum”.

Our findings show that this minimum may not be less than 3α− 2.

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###############

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Figure: G is a well-covered graph with of order |V (G )| = 12 = 4α (G ).


Very well-Covered Graphs

Definition (O. Favaron, Discrete Mathematics 1982)

If G is well-covered, has no isolated vertices, and |V (G )| = 2α (G ),

then G is a very well-covered graph.

The only very well-covered cycle is C4.

Theorem (Levit and Mandrescu, Congressus Numerantium 2007)A connected graph G of girth ≥ 5 is very well-covered iff its pendantedges form a perfect matching, i.e., G = H ◦K1 for some graph H.

w w ww w

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G1@@@

w www

w w w ww w

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G2 w w ww w w

G3 w w wFigure: G1 is well-cov but not very well-cov, G2 = C5 ◦K1 and G3 = P3 ◦K1


Theorem (Levit and Mandrescu, in Graph Theory in Paris 2006)

If G is a very well-covered graph with α(G ) = α,

then s0 ≤ s1 ≤ · · · ≤ sdα/2e and sd(2α−1)/3e ≥ · · · ≥ sα−1 ≥ sα.

ExampleG is very well-covered with α = 10, and satisfies:

s0 = 1 ≤ s1 ≤ s2 ≤ s3 ≤ s4 ≤ s5 = 4920 = sdα/2e and

sd(2α−1)/3e= s7 = 5696 ≥ s8 ≥ s9 ≥ s10 = 144 = sα

as I (G ) = 1+ 10x + 171x2 + 824x3 + 2485x4 + 4920x5 + 6505x6+

+5696x7 + 3174x8 + 1040x9 + 144x10

w w w w w w w w w ww w w w w w w w w w

G

Figure: G is very well-covered with α (G ) = 10Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 22 / 50

Our Refinement

Theorem (Roller-Coaster - Cutler & Pebody, J. Comb Th A 2017)

For well-covered graphs, the part(sd α

2 e, sd α2 e+1, ..., sα

)of

the independence sequence is unconstrained in the sense of Alavi et al.


If G is a very well-covered graph of order ≥ 2 with α(G ) = α,

then s0≤ s1≤ · · · ≤ sdα/2e and sd(2α−1)/3e ≥ · · · ≥ sα−1 ≥ sα.

Conjecture

For every permutation σ of the set {⌈

α2

⌉,⌈

α2

⌉+ 1, ...,

⌈ 2α−13

⌉}, there is

a very well-covered graph G, having α(G ) = α, such that its

independence sequence satisfies sσ(d α2 e) < sσ(d α

2 e+1) < · · · < sσ(d 2α−13 e).


Our Refinement



2 e, sd α2 e+1, ..., sα

)of



If G is a very well-covered graph of order ≥ 2 with α(G ) = α,

then s0≤ s1≤ · · · ≤ sdα/2e and sd(2α−1)/3e ≥ · · · ≥ sα−1 ≥ sα.

Conjecture

For very well-covered graphs, the part (sd α2 e, sd α

2 e+1, ..., sd 2α−13 e) of



1-Well-Covered Graphs

Definition (J. W. Staples, Ph.D. Thesis, 1975)

A well-covered graph G (with at least two vertices) is 1-well-covered

if G − v is well-covered for every vertex v .

The only 1-well-covered cycles are C3 and C5.

P4 is very well-covered, but NOT 1-well-covered.K2 is both very well-covered and 1-well-covered.

ExampleG1 is well-covered; G2 is very well-covered; only G3 is 1-well-covered.

w w w ww w

��

G1

wv w w ww w wG2

wuw w w ww w

�� @

@@

G3

Figure: G1 − v is not well-covered, G2 − u is not well-covered


DefinitionA graph G ∈ W2 if |V (G )| ≥ 2 and every 2 disjoint independent setsare contained in 2 disjoint maximum independent sets.

A well-covered graph G (with at least two vertices) is1-well-covered if G − v is well-covered for every vertex v .

Theorem (J. W. Staples, Ph.D. Thesis, 1975)Let G be a graph without isolated vertices.

Then G is 1-well-covered if and only if G ∈ W2.

A graph with isolated vertices may be 1-well-covered but not in W2.

ExamplesG is 1-well-covered, but G /∈ W2. H is both 1-well-covered & H ∈ W2.

w ww

w ��

G w w w ww w w��@@@

@@@

H


K2 is 1-well-covered and I (K2; x) = 1+ 2x is unimodal.

TheoremIf G is connected and 1-well-covered, α = α (G ), and (sk ) are

the coeffi cients of I (G ; x), then the following are true:

(i)s0(α0

) ≤ s1

2 ·(

α1

) ≤ s2

22 ·(

α2

) ≤ · · · ≤ sα

2α ·(

αα

) ; (∗)

(ii) s0 ≤ s1 ≤ · · · ≤ st where t =⌊2α− 13

⌋.


TheoremIf G is a connected 1-well-covered graph, with α (G ) = α,

then s0 ≤ s1 ≤ · · · ≤ st , where t =⌊ 2α−1

3

⌋.

ExampleG is 1-well-covered with α = 8, and satisfies:

s0 = 1 ≤ s1 ≤ s2 ≤ s3 ≤ s4 = 4825 = sdα/2e ≤ s5= 10472 = sb 2α−13 c as

I (G ; x) = 1+ 24x + 245x2 + 1392x3 + 4825x4 + 10472x5+

+13928x6 + 10400x7 + 3344x8.

w w w w w w w ww w w w w w w ww w w w w w w w

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G

Figure: G is 1-well-covered with α (G ) = 8Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 28 / 50

Our Refinement



2 e, sd α2 e+1, ..., sα

)of




3

⌋.

Conjecture

For every permutation σ of the set {⌈ 2α−1

3

⌉,⌈

α2

⌉+ 1, ..., dαe},

there exists a connected 1-well-covered graph G having α(G ) = α,

such that sσ(d(2α−1)/3e)≤ sσ(d(2α−1)/3e+1)≤ · · · ≤ sσ(α).


Our Refinement



2 e, sd α2 e+1, ..., sα

)of




3

⌋.

Conjecture

For 1-well-covered graphs, the part (sd(2α−1)/3e, sd(2α−1)/3e+1, · · · , sα)

of the independence sequence is unconstrained in the sense of Alavi et al.


Quasi-regularizable graphs

G is a λ-quasi-regularizable graph if λ > 0 and λ · |S | ≤ |N (S)|is true for every independent set S of G .If λ = 1, then G is a quasi-regularizable graph (C. Berge, 1982).

Theorem (C. Berge, 1982)Every well-covered graph is quasi-regularizable.

Examples (of non-well-covered graphs)P5 is not quasi-regularizable

G1 is quasi-regularizable, while G2 is 2-quasi-regularizable

w wa1 a2ww w

P5 w w ww w w��G1 wwv1

v2

w w w ww w

��@

@@

@@@

G2 wuFigure: {a1, a2} , {v1, v2} , {u} are maximal but non-maximum independent sets


Quasi-regularizable graphs

TheoremIf G is a λ-quasi-regularizable graph of order n, then :

sr ≥ sr+1 ≥ · · · ≥ sα(G ), for r =⌈ n−1

λ+2

⌉.

Examples

I (G1; x) = 1+ 7x + 12x2 + 5x3 + 1x4 has s2 = 12 ≥ 5 ≥ 1, as r = 2

I (G2; x) = 1+ 9x + 24x2 + 22x3 + 7x4 + 1x5

has s3 = 22 ≥ 7 ≥ 1, as r = 3

w w w ww w w��

G1 w w w w ww w w w

��

G2

Figure: Quasi-regularizable graphs.


TheoremIf G is a λ-quasi-regularizable graph of order n, then :

sr ≥ sr+1 ≥ · · · ≥ sα(G ), for r =⌈ n−1

λ+2

⌉.

λ = 1 means that G is a quasi-regularizable graph (Berge, 1982).

Well-covered graphs without isolated vertices are quasi-regularizable(Berge, 1982).

CorollaryIf G is a well-covered graph without isolated vertices, then


{α,⌈ n−1

3

⌉}, where |V (G )| = n and α(G ) = α.

Notice that min{

α,⌈ n−1

3



Summary

The "any-order" part of the independence sequence of

a general graph is (s1, s2, ..., sα)

(Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös, 1987);

a well-covered graph is(sd α

2 e, sd α2 e+1, ..., sα

)(J. Cutler and L. Pebody, 2017);

a well-covered graph of order n ≤ 3α− 2 is(sd α

2 e, sd α2 e+1, ..., sd n−13 e

); [OPEN]

a very well-covered graph is(sd α

2 e, sd α2 e+1, ..., sd 2α−1

3 e); [OPEN]

a 1-well-covered graph G is(sd 2α−1

3 e, sd α2 e+1, ..., sα

). [OPEN]


The Best Time to Validate a Conjecture

“The best time to plant a tree was 20 years ago.

The second best time is now.”—Chinese Proverb

Conjecture (Y. Alavi, P. Malde, A. Schwenk and P. Erdös, 1987) :

The independence polynomial of every tree is unimodal.

“The best time to prove the above conjecture was 30 years ago.

The second best time is now.”

Thank you for your attention !


The Best Time to Prove a Theorem

“The best time to plant a tree was 20 years ago.

The second best time is now.”—Chinese Proverb

Conjecture (Y. Alavi, P. Malde, A. Schwenk and P. Erdös, 1987) :

The independence polynomial of every tree is unimodal.

“The best time to prove the above conjecture was 30 years ago.

The second best time is now.”

Thank you for your attention !


Theorem (Alavi, Malde, Schwenk and Erdös, 1987)

For every permutation π of {1, 2, ..., α}, there is a graph Gwith α(G ) = α, such that sπ(1) < sπ(2) < ... < sπ(α).


Ex-Conjecture: I (G ; x) is unimodal for every bipartite graph G .(Levit and Mandrescu, 2006).

D. Galvin, 2012I (G ; x) is unimodal for almost every bipartite graph G .

A. Bhattacharyya and J. Kahn, 2013There is a bipartite graph G whose I (G ; x) is not unimodal.

A. Shwenk, 2015There is a smaller bipartite graph G whose I (G ; x) is not unimodal.


Theorem (J. W. Staples, PhD Thesis, 1975)A graph G without isolated vertices is 1-well-covered ⇔ G ∈ W2 ⇔⇔ α(G − v) = α(G ) and G − v is well-covered, for every v ∈ V (G ).

TheoremFor a connected graph G of order ≥ 2 the following are equivalent:

(i) G is 1-well-covered;

(ii) G 6= P3 and G − v is well-covered, for every v ∈ V (G );

(iii) for each non-maximum indep set A there are at least two disjoint

indep sets B1,B2 such that A∪ B1,A∪ B2 are maximum indep sets in G.

w w wP3v v v

w w ww w wH

wuw w w ww w

�� @

@@

G

Figure: P3 − v and G − v is well-covered for every vertex v


Fact (J. W. Staples, Ph.D. Thesis, 1975)

If G 6= K2 is 1-well-covered, then G has no leaf.

TheoremIf G 6= K2 is connected and 1-well-covered, then the following are true:(i) for each v ∈ V (G ), there are at least two disjoint maximumindependent sets S1, S2 in G such that v /∈ S1 ∪ S2;

(ii) G has at least 2α(G ) + 1 vertices.

Proof.Let v ∈ V (G ) and u,w ∈ N(v). (By Fact, |N(v)| ≥ 2).G 6= K2 is 1-well-covered ⇔ G ∈W2

G ∈W2 ⇒ are two disjoint maximum independent sets S1,S2 in Gsuch that u ∈ S1,w ∈ S2.u,w ∈ N(v) ⇒ vu, vw ∈ E (G ) ⇒ v /∈ S1 ∪ S2 ⇒ (i).Consequently, 1+ 2α (G ) = |{v} ∪ S1 ∪ S2| ≤ |V (G )| ⇒ (ii).


Theorem (L & M —2016)For a graph G of order two at least, and having no isolated vertices,the following assertions are equivalent:(i) G is in the class W2;(ii) G is 1-well-covered;(iii) α(G − v) = α(G ) and G − v is well-covered, for every v ∈ V (G );(iv) G 6= P3 and G − v is well-covered, for every v ∈ V (G );(v) for each non-maximum independent set A in G there are at least

two disjoint independent sets B1,B2 such that A∪ B1,A∪ B2are maximum independent sets in G;

(vi) for every non-maximum independent set A in G there are at leasttwo different independent sets B1,B2 such that A∪ B1,A∪ B2are maximum independent sets in G;

(vii) for each non-maximum independent set A in G and v /∈ A, there issome maximum independent set S such that A ⊂ S and v /∈ S.

(i) ⇔ (ii) ⇔ (iii) proved by J. W. Staples, PhD Thesis, 1975


Questions and facts on I (G ; x) = s0 + s1x + s2x2 + ...+ sα(G )xα(G )

Is I (G ; x) log-concave, i.e., s2i ≥ si−1 · si+1 for 1 ≤ i < α(G )?

Folklore: If I (G ; x) is log-concave, then I (G ; x) is also unimodal.

Some results:

(i) I (G ; x) is log-concave, whenever all its roots are real (I. Newton).

(ii) I (G ; x) is log-concave for any claw-free graph G

(the "claw" is K1,3); (Y. O. Hamidoune, 1990)

(iii) I (G ◦K1; x) is log-concave for every graph G with α(G ) ≤ 3(Levit and Mandrescu, 2004)

(iv) I (K1,n ◦K1; x) is log-concave for every n-star K1,n(Levit and Mandrescu, 2004)



Are all the roots of I (G ; x) real?

Some results:

(i) a root of smallest modulus of I (G ; x) is real(J. Brown, K. Dilcher, R. Nowakowski, J. Algebr. Comb. 2000).

(ii) if L(G ) is the line graph of G , then I (L(G ); x) has all its rootsreal, since I (L(G ); x) = M(G ; x) and M(G ; x) has only real roots.

(iii) if G is claw-free, then I (G ; x) has all its roots real(M. Chudnovsky, P. Seymour, J. Comb. Th. 2007).

(iv) I (G ; x) has only real roots iff I (G ◦K1; x) has all the roots real,(Levit and Mandrescu, DAM 2008).

(v) All the real roots of I (G ◦Kp ; x) belong to[− 1p ,−

1n(p+1)

),

where n = |V (G )|.(Mandrescu, Graphs and Combin. 2009).



Are there closed formulae for I (G ; x) ?

Some results:(i) If Pn denotes the chordless path on n ≥ 3, then

I (Pn; x) =b(n+1)/2c

∑j=0

(n+1−jj ) · x j .

(J. L. Arocha, 1984).(ii) If Cn denotes the chordless cycle on n ≥ 3, then

I (Cn; x) =bn/2c

∑j=0

nn−j · (

n−jj ) · x j

(F. Harary and I. Gutman, 1983).(iii) if Sn denotes the graph obtained from K1,n by appendinga single pendant edge to each vertex of K1,n, then

I (Sn; x) = (1+ x) ·n∑k=0

[(nk) · 2k + (

n−1k−1)

]· xk ,

(Levit and Mandrescu, 2008)Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 43 / 50

Theorem (Levit and Mandrescu, DAM 2016)

Let G be with E (G ) 6= ∅. If I (G ◦H; x) has only real roots,then the same is true for both I (G ; x) and I (H; x).

The converse is not necessarily true; e.g.,

I (K2; x) = 1+ 2x and I (3K1; x) = (1+ x)3 have only real roots,

while I (K2 ◦ 3K1; x) = 1+ 8x + 21x2 + 26x3 + 17x4 + 6x5 + x6

has non-real roots. Notice that: H = 3K1 has α (H) = 3.

w wK2

HHHH

HH

@@@

��

��

��

K2 ◦ 3K1

w w w3K1

w w w w w ww w

Theorem (Mandrescu, G&C 2009; Levit & Mandrescu, DAM 2016)

Let G and H be graphs with E (G ) 6= ∅ and α (H) ≤ 2.Then I (G ◦H; x) has only real roots iff I (G ; x) has only real roots.


Theorem (E. Mandrescu, Graphs & Combin 2009)Let G be a connected graph on n ≥ 2 vertices. Then the following hold:(i) −1/p is a root of I (G ◦Kp ; x) with the multiplicity n− α(G ) ≥ 1;(ii) there is a bijection between the set of roots of I (G ◦Kp ; x) differentfrom −1/p and the set of roots of I (G ; x), respecting the multiplicitiesof the roots; moreover, real roots correspond to real roots.

A graph is called claw-free if it has no subgraph isomorphic to K1,3.

Theorem (M. Chudnovsky and P. Seymour, J Combin Th 2007 )The independence polynomial of a claw-free graph has only real roots.

CorollaryLet G be a connected graph on n ≥ 2 vertices, and p ≥ 2. Then(i) G ◦Kp is 1-well-covered (⇔ G ◦Kp ∈ W2);

(ii) if I (G ; x) has only real roots (e.g., G is claw-free), thenI (G ◦Kp ; x) has only real roots, and hence it is unimodal.


Corona of graphs

Definition (R. Frucht, F. Harary, Aequationes Math. 1970)

The corona G ◦ H of G and the family H = {Hv : v ∈ V (G )} isthe disjoint union of G and Hv , v ∈ V (G ), with additional edgesjoining each vertex v ∈ V (G ) to all the vertices of Hv .

If all Hv = H, then we denote G ◦H instead of G ◦ H.

��

��

��

��

��

��

AAA

AAA

AAA

AAA

AAA

AAA

w w w w w wr r r r r r r r r r r r r r r r r r r rv1 v2 vi−1 vi vi+1 vn

Km Km Km Km Km Km

Figure: The corona Pn ◦Km .Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 46 / 50

Corona of graphs

Definition (R. Frucht, F. Harary, Aequationes Math. 1970)

The corona G ◦ H of G and the family H = {Hv : v ∈ V (G )} isthe disjoint union of G and Hv , v ∈ V (G ), with additional edgesjoining each vertex v ∈ V (G ) to all the vertices of Hv .

If all Hv = H, then we denote G ◦H instead of G ◦ H.

��

��

��

��

��

��

AAA

AAA

AAA

AAA

AAA

AAA

w w w w w wr r r r r r r r r r r r r r r r r r r rv1 v2 vi−1 vi vi+1 vn

H1 H2 Hi−1 Hi Hi+1 Hn

Figure: The corona Pn ◦ (H1,H2, ...,Hn).Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 47 / 50

Theorem (J. Topp, L. Volkman, Ars Comb. 1992)

The corona G ◦ H of G and H = {Hv : v ∈ V (G )} is well-covered

if and only if each Hv ∈ H is a complete graph on at least one vertex.

Theorem (L & M —2016. For G without isolated vertices.)The corona G ◦ H of G and H = {Hv : v ∈ V (G )} is 1-well-covered

if and only if each Hv ∈ H is a complete graph on at least two vertices.

ExampleG1 = P2 ◦ {K1, 2K1}, G2 = P2 ◦ {K1,K2}, G3 = P2 ◦ {K2,K3}

��

w w ww wG1

w��

w w www wG2

@@@ �

��

��

��w w w w ww ww wG3

Figure: G1 is not well-cov, G2 is well-cov, while G3 is 1-well-covered.


Basic questions about independence polynomials

Which necessary / suffi cient conditions on G guarantee that I (G ; x)

has only real roots ?

is log-concave ?

is unimodal ?

Which graph operations lead to graphs whose independence polynomials

have only real roots ?

are log-concave ?

are unimodal ?


Class W2

Definition (J. W. Staples, Ph.D. Thesis, 1975)

A graph G belongs to class W2 if |V (G )| ≥ 2 and every 2 disjointindependent sets are contained in 2 disjoint maximum independent sets.

Every complete graph Kn ∈ W2 for n ≥ 2.K2 is both very well-covered and in class W2.

W1 consists of all well-covered graphs. Clearly, W2 ⊂ W1.

ExampleG1 is well-covered; G2 is very well-covered; only G3 is in class W2.

w w w ww w

��

G1 a b

wv w w ww w wG2

wva b w w w w

w w�� @

@@

G3

Figure: G1 and G2 are not in W2 because {v} and {a, b} ...Vadim E. Levit Ariel University, Israel &, Eugen Mandrescu Holon Institute of Technology, Israel ()Roller-Coaster Conjecture 25/6/17 50 / 50

a re–ned version of the roller-coaster conjecture mandrescu.pdf · independent sets and...

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