The asymptotic value of the independence ratio for the direct graph power Ágnes Tóth

The asymptotic value of the independence ratio for the direct graph power

Ágnes Tóth

Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences

CanaDAM 2013

The asymptotic value of the independence ratio for the direct graph power

independence ratio of a graph G: i(G) = _|V^α(₍^G_G⁾_)|

direct product of two graphs G and H:

the graph G×H for which V(G×H) =V(G)×V(H), and {(x1,y1),(x2,y2)} ∈E(G×H), i

{x1,x2} ∈E(G)and{y1,y2} ∈E(H)}.

G

H G×H

G^×k denotes the kth direct power of G

Denition (Brown, Nowakowski, Rall - 1996.):

The asymptotic value of the independence ratio for the direct graph power is dened as

A(G) = lim

k→∞i(G^×k).

The asymptotic value of the independence ratio for the direct graph power

independence ratio of a graph G: i(G) = _|V^α(₍^G_G⁾_)|

direct product of two graphs G and H:

the graph G×H for which V(G×H) =V(G)×V(H), and {(x1,y1),(x2,y2)} ∈E(G×H), i

{x1,x2} ∈E(G)and{y1,y2} ∈E(H)}.

G

H G×H

G^×k denotes the kth direct power of G

Denition (Brown, Nowakowski, Rall - 1996.):

The asymptotic value of the independence ratio for the direct graph power is dened as

A(G) = lim

k→∞i(G^×k).

The asymptotic value of the independence ratio for the direct graph power

independence ratio of a graph G: i(G) = _|V^α(₍^G_G⁾_)|

direct product of two graphs G and H:

the graph G×H for which V(G×H) =V(G)×V(H), and {(x1,y1),(x2,y2)} ∈E(G×H), i

{x1,x2} ∈E(G)and{y1,y2} ∈E(H)}.

G

H G×H

G^×k denotes the kth direct power of G

Denition (Brown, Nowakowski, Rall - 1996.):

The asymptotic value of the independence ratio for the direct graph power is dened as

A(G) = lim

k→∞i(G^×k).

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U) UNG(U)

U N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U) UNG(U)

U N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U

NG(U) UNG(U)

U N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U)

UNG(U) U N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U)

U NG(U)

U N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U) UNG(U)

U N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U) UNG(U)

U

N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U) UNG(U)

U N^G(U)

there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

U NG(U) UNG(U)

U N^G(U) there exists an independent set U_k of G^×k such that

|Uk|

|U_k|+|N_G×k(U_k)| ≥ _|U|+|^|U_NG^|_(U)|

and

klim→∞

|Uk|

|V(G^×k)| = _|U|+|^|_N^U|

G(U)|

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Theorem (BNR): If A(G)> ¹₂, then A(G) =1.

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Theorem (BNR): If A(G)> ¹₂, then A(G) =1.

Example: bipartite graphs

for bipartite G we have i(G)≥ ¹₂ and so A(G)≥ ¹₂ ifα(G)> ¹₂|V(G)|then A(G) =1

ifα(G) = ¹₂|V(G)|then G has a perfect matching, therefore G^×k also has one (∀k)

and i(G^×k)≤ ¹₂ thus A(G) = ¹_{2 G}

G G^×2

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Theorem (BNR): If A(G)> ¹₂, then A(G) =1.

Example: bipartite graphs

for bipartite G we have i(G)≥ ¹₂ and so A(G)≥ ¹₂

ifα(G)> ¹₂|V(G)|then A(G) =1

ifα(G) = ¹₂|V(G)|then G has a perfect matching, therefore G^×k also has one (∀k)

and i(G^×k)≤ ¹₂ thus A(G) = ¹_{2 G}

G G^×2

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Theorem (BNR): If A(G)> ¹₂, then A(G) =1.

Example: bipartite graphs

for bipartite G we have i(G)≥ ¹₂ and so A(G)≥ ¹₂ if α(G)> ¹₂|V(G)|then A(G) =1

ifα(G) = ¹₂|V(G)|then G has a perfect matching, therefore G^×k also has one (∀k) and i(G^×k)≤ ¹₂ thus A(G) = ¹₂

G

G G^×2

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Theorem (BNR): If A(G)> ¹₂, then A(G) =1.

Example: bipartite graphs

for bipartite G we have i(G)≥ ¹₂ and so A(G)≥ ¹₂ if α(G)> ¹₂|V(G)|then A(G) =1

if α(G) = ¹₂|V(G)|then G has a perfect matching, therefore G^×k also has one (∀k) and i(G^×k)≤ ¹₂ thus A(G) = ¹₂

G

G G^×2

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Theorem (BNR): If A(G)> ¹₂, then A(G) =1.

Results of Brown, Nowakowski and Rall

0<i(G)≤i(G^×2)≤i(G^×3)≤ · · · ≤A(G)≤1 Theorem (Brown, Nowakowski, Rall - 1996.):

For any independent set U of G we have A(G)≥ |U|+|^|N^UG^|(U)|, where N_G(U) denotes the neighbourhood of U in G.

Theorem (BNR): If A(G)> ¹₂, then A(G) =1.

Observation (Alon, Lubetzky): A(G)≥i_max^∗ (G), where i_max(G) = max

U independent in G

|U|

|U|+|NG(U)|

i_max^∗ (G) =

(i_max(G), if i_max(G)≤ ¹₂ 1, if imax(G)> ¹₂.

Questions of Alon and Lubetzky

i(G)^∃G≤^:<i_max(G)^∃G≤^:<i_max^∗ (G)≤A(G) Question (Alon, Lubetzky - 2007.):

Does every graph G satisfy A(G) =i_max^∗ (G)?

Theorem (Á. Tóth - 2012.): A(G) =i_max^∗ (G), for any graph G. It easily follows from the inequality

i_max^∗ (G ×H)≤max{i_max^∗ (G),i_max^∗ (H)}.

Proposition (weaker inequality): i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}

Questions of Alon and Lubetzky

i(G)^∃G≤^:<i_max(G)^∃G≤^:<i_max^∗ (G)≤A(G) Question (Alon, Lubetzky - 2007.):

Does every graph G satisfy A(G) =i_max^∗ (G)?

Theorem (Á. Tóth - 2012.): A(G) =i_max^∗ (G), for any graph G.

It easily follows from the inequality

i_max^∗ (G ×H)≤max{i_max^∗ (G),i_max^∗ (H)}.

Proposition (weaker inequality): i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}

Questions of Alon and Lubetzky

i(G)^∃G≤^:<i_max(G)^∃G≤^:<i_max^∗ (G)≤A(G) Question (Alon, Lubetzky - 2007.):

Does every graph G satisfy A(G) =i_max^∗ (G)?

Theorem (Á. Tóth - 2012.): A(G) =i_max^∗ (G), for any graph G.

It easily follows from the inequality

i_max^∗ (G ×H)≤max{i_max^∗ (G),i_max^∗ (H)}.

Proposition (weaker inequality): i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}

Questions of Alon and Lubetzky

i(G)^∃G≤^:<i_max(G)^∃G≤^:<i_max^∗ (G)≤A(G) Question (Alon, Lubetzky - 2007.):

Does every graph G satisfy A(G) =i_max^∗ (G)?

Theorem (Á. Tóth - 2012.): A(G) =i_max^∗ (G), for any graph G.

It easily follows from the inequality

i_max^∗ (G ×H)≤max{i_max^∗ (G),i_max^∗ (H)}.

Proposition (weaker inequality): i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}

Consequences

Conjecture (BNR): A(G ∪H) =max{A(G),A(H)}, where A∪G denotes the disjoint union of G and H.

Theorem (BNR):

For any rational r∈(0,¹₂]∪{1}there exists a graph G with A(G) =r. Question (BNR): Can the value of A(G) be irrational?

From A(G) =i_max^∗ (G) we obtain that: A(G∪H) =max{A(G),A(H)}. A(G) cannot be irrational.

Consequences

Conjecture (BNR): A(G ∪H) =max{A(G),A(H)}, where A∪G denotes the disjoint union of G and H.

Theorem (BNR):

For any rational r∈(0,¹₂]∪{1}there exists a graph G with A(G) =r.

Question (BNR): Can the value of A(G) be irrational?

From A(G) =i_max^∗ (G) we obtain that: A(G∪H) =max{A(G),A(H)}. A(G) cannot be irrational.

Consequences

Conjecture (BNR): A(G ∪H) =max{A(G),A(H)}, where A∪G denotes the disjoint union of G and H.

Theorem (BNR):

For any rational r∈(0,¹₂]∪{1}there exists a graph G with A(G) =r.

Question (BNR): Can the value of A(G) be irrational?

From A(G) =i_max^∗ (G) we obtain that:

A(G∪H) =max{A(G),A(H)}.

A(G) cannot be irrational.

Algorithmic aspects

Question (BNR): Is A(G) computable?

And if so, what is its complexity?

Theorem (BNR):

If G is bipartite then A(G) can be determined in polynomial time. Theorem (AL):

Determining whether A(G) = 1 or A(G) ≤ ¹₂ can be also done in polynomial time.

From A(G) =i_max^∗ (G) we also obtain that:

The problem of deciding whether A(G)>t for a given graph G and a value t, is NP-complete.

Algorithmic aspects

Question (BNR): Is A(G) computable?

And if so, what is its complexity?

Theorem (BNR):

If G is bipartite then A(G) can be determined in polynomial time.

Theorem (AL):

Determining whether A(G) = 1 or A(G) ≤ ¹₂ can be also done in polynomial time.

From A(G) =i_max^∗ (G) we also obtain that:

The problem of deciding whether A(G)>t for a given graph G and a value t, is NP-complete.

Algorithmic aspects

Question (BNR): Is A(G) computable?

And if so, what is its complexity?

Theorem (BNR):

If G is bipartite then A(G) can be determined in polynomial time.

Theorem (AL):

Determining whether A(G) = 1 or A(G) ≤ ¹₂ can be also done in polynomial time.

From A(G) =i_max^∗ (G) we also obtain that:

The problem of deciding whether A(G)>t for a given graph G and a value t, is NP-complete.

Algorithmic aspects

Question (BNR): Is A(G) computable?

And if so, what is its complexity?

Theorem (BNR):

If G is bipartite then A(G) can be determined in polynomial time.

Theorem (AL):

Determining whether A(G) = 1 or A(G) ≤ ¹₂ can be also done in polynomial time.

From A(G) =i_max^∗ (G) we also obtain that:

The problem of deciding whether A(G)>t for a given graph G and a value t, is NP-complete.

The Hedetniemi conjecture

Hedetniemi's conjecture - 1966.:

For every graph G and H we have χ(G×H) =min{χ(G), χ(H)}.

G

H G×H

The fractional version of the conjecture:

(χ_f denotes the fractional chromatic number of the graph.)

χ_f(G×H) =min{χ_f(G), χ_f(H)}. χ_f(G ×H)≤min{χ_f(G), χ_f(H)}is easy.

Tardif, 2005.:χ_f(G ×H)≥ ¹₄min{χ_f(G), χ_f(H)}. Theorem (Zhu - 2010.):

The fractional version of Hedetniemi's conjecture is true. Corollary: The Burr-Erd®s-Lovász conjecture is true.

The Hedetniemi conjecture

Hedetniemi's conjecture - 1966.:

For every graph G and H we have χ(G×H) =min{χ(G), χ(H)}.

G

H G×H

The fractional version of the conjecture:

(χ_f denotes the fractional chromatic number of the graph.)

χ_f(G×H) =min{χ_f(G), χ_f(H)}. χ_f(G ×H)≤min{χ_f(G), χ_f(H)}is easy.

Tardif, 2005.:χ_f(G ×H)≥ ¹₄min{χ_f(G), χ_f(H)}. Theorem (Zhu - 2010.):

The fractional version of Hedetniemi's conjecture is true. Corollary: The Burr-Erd®s-Lovász conjecture is true.

The Hedetniemi conjecture

Hedetniemi's conjecture - 1966.:

For every graph G and H we have χ(G×H) =min{χ(G), χ(H)}.

G

H G×H

The fractional version of the conjecture:

(χ_f denotes the fractional chromatic number of the graph.)

χ_f(G×H) =min{χ_f(G), χ_f(H)}. χ_f(G ×H)≤min{χ_f(G), χ_f(H)}is easy.

Tardif, 2005.:χ_f(G ×H)≥ ¹₄min{χ_f(G), χ_f(H)}.

Theorem (Zhu - 2010.):

The fractional version of Hedetniemi's conjecture is true. Corollary: The Burr-Erd®s-Lovász conjecture is true.

The Hedetniemi conjecture

Hedetniemi's conjecture - 1966.:

For every graph G and H we have χ(G×H) =min{χ(G), χ(H)}.

G

H G×H

The fractional version of the conjecture:

(χ_f denotes the fractional chromatic number of the graph.)

χ_f(G×H) =min{χ_f(G), χ_f(H)}. χ_f(G ×H)≤min{χ_f(G), χ_f(H)}is easy.

Tardif, 2005.:χ_f(G ×H)≥ ¹₄min{χ_f(G), χ_f(H)}.

Theorem (Zhu - 2010.):

The fractional version of Hedetniemi's conjecture is true.

Corollary: The Burr-Erd®s-Lovász conjecture is true.

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H; furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H; furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H; furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H; furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

y

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G,

for∀x∈V(G)the projection of the x-slice of B is independent in H; furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G,

for∀x∈V(G)the projection of the x-slice of B is independent in H; furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

x

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

y

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

y

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

MA={(x,y)∈V(G×H) :

∃(x⁰,y)∈A,{x,x⁰} ∈E(G)}

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

x

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

x

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A,

andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

MB ={(x,y)∈V(G ×H) :

∃(x,y⁰)∈B,{y,y⁰} ∈E(H)}

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - Zhu's lemma

G H

any independent set U of G×H

can be partitioned into the union ofAandB, where

for∀y∈V(H)the projection of the y-slice of A is independent in G, for∀x∈V(G)the projection of the x-slice of B is independent in H;

furthermore ifMAdenotes the G-neighbourhood of A, andMBdenotes the H-neighbourhood of B,

A, B, MA, MB are pairwise disjoint subsets of V(G×H).

The idea of the proof - proof of the weaker proposition

Zhu's lemma⇒ i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}:

i(G ×H) = _|V^α(_(G^G_×⁾_H)| = _|V(G^|U_×^|_H)|

|A|

|A|+|MA| ≤i_max(G),

|B|

|B|+|MB| ≤i_max(H)

|A|+|B|=|U|,|A|+|B|+|MA|+|MB| ≤ |V(G×H)|

The idea of the proof - proof of the weaker proposition

Zhu's lemma⇒ i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}:

i(G ×H) = _|V^α(_(G^G_×⁾_H)| = _|V(G^|U_×^|_H)|

|A|

|A|+|MA| ≤i_max(G),

|B|

|B|+|MB| ≤i_max(H)

|A|+|B|=|U|,|A|+|B|+|MA|+|MB| ≤ |V(G×H)|

The idea of the proof - proof of the weaker proposition

Zhu's lemma⇒ i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}:

i(G ×H) = _|V^α(_(G^G_×⁾_H)| = _|V(G^|U_×^|_H)|

|A|

|A|+|MA| ≤i_max(G),

|B|

|B|+|MB| ≤i_max(H)

|A|+|B|=|U|,|A|+|B|+|MA|+|MB| ≤ |V(G×H)|

G H

The idea of the proof - proof of the weaker proposition

Zhu's lemma⇒ i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}:

i(G ×H) = _|V^α(_(G^G_×⁾_H)| = _|V(G^|U_×^|_H)|

|A|

|A|+|MA| ≤i_max(G),

|B|

|B|+|MB| ≤i_max(H)

|A|+|B|=|U|,|A|+|B|+|MA|+|MB| ≤ |V(G×H)|

G H

y

The idea of the proof - proof of the weaker proposition

Zhu's lemma⇒ i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}:

i(G ×H) = _|V^α(_(G^G_×⁾_H)| = _|V(G^|U_×^|_H)|

|A|

|A|+|MA| ≤i_max(G), _|B|+|^|B_MB^| _| ≤i_max(H)

|A|+|B|=|U|,|A|+|B|+|MA|+|MB| ≤ |V(G×H)|

G H

x

The idea of the proof - proof of the weaker proposition

Zhu's lemma⇒ i(G×H)≤max{i_max^∗ (G),i_max^∗ (H)}:

i(G ×H) = _|V^α(_(G^G_×⁾_H)| = _|V(G^|U_×^|_H)|

|A|

|A|+|MA| ≤i_max(G), _|B|+|^|B_MB^| _| ≤i_max(H)

|A|+|B|=|U|,|A|+|B|+|MA|+|MB| ≤ |V(G ×H)|

G H

Thank you for your attention!