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Don’t be dense, try hypergraphs!

Anthony BonatoRyerson University

Graphs @ Ryerson

Independent sets

• set of vertices in an undirected, simple graph, no two of which are adjacent

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Paths

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…• number of

independent sets = F(n+2)

- Fibonacci numberPn

Stars

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• number of independent sets = 1+2n

• independence density

= 2-n-1+½

K1,n

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Independence density

• G a graph of order n

• i(G) = number of independent sets in G (including ∅)– Fibonacci number of G

• id(G) = i(G) / 2n

– independence density of G– rational in (0,1]

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Properties

• if G is a spanning subgraph of H, then

i(H) ≤ i(G)– (possibly) fewer edges in G

• i(G U H) = i(G)i(H)

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Monotonicity

• if G is subgraph of H, then id(H) ≤ id(G)– G has an edge, then: id(G) ≤ id(K2) = ¾

Proof: Say G has order m and H has order n.

id(H) = i(H) / 2n

≤ i(G U (H-G)) / 2n (disjoint union)= i(G)i(H-G) / 2m2n-m

= id(G)id(H-G)

≤ id(G).

• id(G U H) = id(G)id(H)

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Infinite graphs?

• view countably a infinite graph as a limit of chains

• extend definition by continuity:

• well-defined?• possible values?

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Chains

• let G be a countably infinite graph

• a chain in G is a set of induced subgraphs Gi such that:– for all i, Gi is an induced subgraph of Gi+1 and

• write id(G, 𝐶) =

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Existence and uniqueness

Theorem (B,Brown,Kemkes,Prałat,11)

Let G be a countably infinite graph.

1. For each chain , id(G, ) exists.

2. For all chains and ’ in G,

id(G, )=id(G, ’).

• proof follows by boundedness and monotonicity

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Examples

• infinite star: id(K1,∞) = 1/2

• one-way infinite path: id(P∞) = 0

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Bounds on id

• if G contains an infinite matching, then id(G) = 0• matching number of G, written µ(G), is the supremum of

the cardinalities of pairwise non-intersecting edges in G

Theorem (B,Brown,Kemkes,Prałat,11)

If µ(G) is finite, then:

• in particular, id(G) = 0 iff µ(G) is infinite

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Rationality

Theorem (BBKP,11)

Let G be a countable graph.

1. id(G) is rational.

2. The closure of the set

{id(G): G countable}

is a subset of the rationals.

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Aside: other densities

• many other density notions for graphs and hypergraphs:– upper density– homomorphism density– Turán density– co-degree density– cop density, …

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Question

• hereditary graph class X: closed under induced subgraphs

• egs: X = independent sets; cliques; triangle-free graphs;

perfect graphs; H-free graphs

• Xd(G) = proportion of subsets which induce a graph in X– generalizes to infinite graphs via chains

• is Xd(G) rational?

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Try hypergraphs!

• hypergraph H = (V,E), E = hyperedges• independent set: does not contain a

hyperedge

• id(H) defined analogously– extend to infinite hypergraphs by continuity– well-defined

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2 3

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Examples

∅,{1},{2},{3},{4},{1,2},{1,3},{2,3},{1,4},{3,4},{1,3,4}

id(H) = 11/16H

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Examples, cont

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…

id(H) = 7/8

Hypergraph id’s

examples:

1. graph, E = subsets of vertices containing a copy of K2

– recovers the independence density of graphs

2. graph, fix a finite graph F; E = subsets of vertices containing a copy of F– F-free density (generalizes (1)).

3. relational structure (graphs, digraphs, orders, etc); F a set of finite structures; E = subsets of vertices containing a member of F – F-free density of a structure (generalizes (2))

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Bounds on id

• matching number of hypergraph H, written µ(H), is the supremum of the cardinalities of pairwise non-intersecting hyperedges in H

Theorem (B,Brown,Mitsche,Prałat,14)

Let H be a hypergraph whose hyperedges have cardinality bounded by k > 0. If µ(H) is finite, then:

• sharp if k = 1,2; not sure lower bound is sharp if k > 2

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Rationality

• rank k hypergraph: hyperedges bounded in cardinality by k > 0– finite rank: rank k for some k

Theorem (BBMP,14): If H has finite rank,

then id(H) is rational.

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Sketch of proof

• notation: for finite disjoint sets of vertices A and B

idA,B(H) = density of independent sets containing A and not B

• analogous properties to id(H) = id ,∅ ∅(H)

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Properties of idA,B(H)

1. For a vertex x outside A U B:

.

2. For a set W outside A U B:

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Out-sets

• for a given A, B, and any hyperedge S such that S∩B = ∅, the set S \ A is the out-set of S relative to A and B – example: A B

S• notation: idr

A,B(H) denotes that every out-set has cardinality at most r

• note that: idk,∅ ∅(H) = id(H)

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Claims

1. If A is not independent, then idrA,B(H) = 0.

2. If A is independent and there is an infinite family of disjoint out-sets, then idr

A,B(H) = 0.

3. If A is independent, then id0A,B(H)=2-(|A|+|B|).

4. Suppose that A is independent and there is no infinite family of disjoint out-sets. If O1, O2,…, Os is a maximal family of disjoint out-sets, then for all r > 0,

where W is the union of the Oi.

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Final steps…

• start with idk,∅ ∅(H)

• by (2), assume wlog there are finitely many out-sets• as A and B are empty, the out-sets are disjoint

hyperedges with union W• by (4):

• apply induction using (1)-(3); each term is rational, or a finite sum of terms where we can apply (1)-(3)

• process ends after k steps

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Unbounded rank

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…

H

Theorem (BBMP,14) 1. The independence density of H is

.

2. The value is S is irrational.

• proof of (2) uses Euler’s Pentagonal Number Theorem

Any real number

• case of finite, but unbounded hyperedges• Hunb = {x: there is a countable hypergraph

H with id(H) = x}

Theorem (BBMP,14) Hunb = [0,1].

• contrasts with rank k case, where there exist gaps such as (1-1/2k,1)

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Independence polynomials

• H finite, independence polynomial of H wrt x > 0

i(H,x) = ,

where ik is the number of independent sets of order k in H

– for eg: id(H) = i(H,1)/2n

• example– i(Pn,x) = i(Pn-1,x) + xi(Pn-2,x),

i(P1,x)=1+x, (P2,x)=1+2x

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Independence densities at x

• (,x): defined in natural way for fixed x ≥ 0– may depend on chain if x ≠ 1– (,1) = id(H)

• examples: – (,x) = 0 for all x

• generalizes to chains with βn = o(n)

– (H,x) = 0 if x < 1:• bounded above by ((1+x)/2)n =o(1)

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Examples, continued

• +

• with chain (Pn: n ≥ 1), we derive that:

(,x) =

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Asymptotic behaviour

Theorem (BBMP,14) For a countable hypergraph which is a limit of the chain

= (Hm: m > 0), we have that

is either 0 or ∞.

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Sketch of proof

• the function is non-decreasing with x, so the limit exists and is in [0,∞]

• for a contradiction, suppose that

= z (0,∞) • choose x0 so that for all x ≥ x0,

z/2 < < 2z • then

> 4

> 2z,

a contradiction.

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Examples, continued

• for each r > 1, can choose chain such that

,x) =

• r is a jumping point• can choose chains where for all x,

,x) is 0, or does not exist

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Future directions

• classify gaps among densities for given hypergraphs– eg: rank k, (1-1/2k,1) is a gap

• rationality of closure of set of id’s for rank k hypergraphs

• which hypergraphs have jumping points, and what are their values?

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General densities

• d: hypergraph function satisfying:– multiplicative on disjoint unions– monotone increasing on subgraphs

• d(H) well-defined for infinite hypergraphs

• properties of d(H)? for eg, when rational?

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