Hypergraphs and hypergraphic polytopes

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Hypergraphs and hypergraphic polytopes

Sophie Rehberg

Doctorants en G´eom´etrie Combinatoire (DGeCo)

June 24, 2021

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Hypergraphs, colorings, orientations Hypergraphic polytopes Generalized permutahedra Reciprocity

Motivation

Stanley (1973): For a graphg,m∈Z>0

χ_g(m) := #properm-colorings ofg is a polynomial inmand

(−1)^dχ_g(−m) = #pairs of compatiblem-colorings

and acyclic orientations.

Theorem (Aguiar, Ardila 2017; Billera, Jia, Reiner 2009) For generalized permutahedraP ⊂R^d,m∈Z>0

χ_d(P)(m) := #

P-generic directionsy ∈(R^d)^∗ with y ∈[m]^d agrees with a polynomial inm of degreed. Moreover,

(−1)^dχ_d(P)(−m) = X

y∈[m]^d

# (vertices of P_y) .

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Motivation

Stanley (1973): For a graphg,m∈Z>0

χ_g(m) := #properm-colorings ofg is a polynomial inmand

(−1)^dχ_g(−m) = #pairs of compatiblem-colorings

and acyclic orientations.

χ_d(P)(m) := #

P-generic directionsy ∈(R^d)^∗ with y ∈[m]^d

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Outline

1 Hypergraphs, colorings, orientations

2 Hypergraphic polytopes

3 Generalized permutahedra

4 Reciprocity

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Hypergraphs and colorings

Ahypergraphh= (I,E) is a pair of

• a finite setI ={n₁, . . . ,n_d}of nodes

• a finite multisetE of non-empty subsets e ⊆I calledhyperedges.

c(ni)∈[m] to every node ni ∈I.

A coloringc :I →[m] is proper if every hyperedgee ∈E contains a unique noden_i ∈e with maximal color.

other notions of proper: not monochromatic, all nodes with different colors within a hyperedge [BDK12, EH66, BTV15, AH05]

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Hypergraphs and colorings

Am-coloring of h= (I,E) is a map c:I →[m] assigning acolor c(ni)∈[m] to every node ni ∈I.

A coloringc :I →[m] is properif every hyperedge e ∈E contains a unique noden_i ∈e with maximal color.

other notions of proper: not monochromatic, all nodes with different colors within a hyperedge [BDK12, EH66, BTV15, AH05]

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Hypergraphs and colorings

Am-coloring of h= (I,E) is a map c:I →[m] assigning acolor c(ni)∈[m] to every node ni ∈I.

A coloringc :I →[m] is properif every hyperedge e ∈E contains a unique noden_i ∈e with maximal color.

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Headings

Aheadingσ ofh = (I,E) is a mapσ:E →I such that fore ∈E we haveσ(e)∈e.

Anoriented cycle in a headingσ ofh is a sequence e₁, . . . ,e_l s.t.

σ(e1)∈e2\σ(e2) σ(e2)∈e3\σ(e3)

...

σ(el)∈e1\σ(e1).

A headingσof a hypergraph h is called acyclic if it does not

Different notions of orientations: [BBM19, Rus13]

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Headings

Aheadingσ ofh = (I,E) is a mapσ:E →I such that fore ∈E we haveσ(e)∈e.

Anoriented cycle in a headingσ ofh is a sequence e₁, . . . ,e_l s.t.

σ(e1)∈e2\σ(e2) σ(e2)∈e3\σ(e3)

...

σ(el)∈e1\σ(e1).

A headingσof a hypergraph h is called acyclic if it does not

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Acyclic headings and compatible colorings

Example:

A coloringc and a headingσ arecompatible if c(σ(e)) = max

j∈e c(j).

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Acyclic headings and compatible colorings

Example:

c(σ(e)) = max

j∈e c(j).

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Acyclic headings and compatible colorings

Example:

A coloringc and a headingσ arecompatible if c(σ(e)) = max

j∈e c(j).

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Reciprocity for hypergraphs

Theorem (Aval, Karaboghossian, Tanasa 2020)

For a hypergraphh = (I,E) with I ={n₁, . . . ,nd},m∈Z>0

χ_d(h)(m) := #(proper colorings of h with m colors) agrees with a polynomial inm of degreed. Moreover,

(−1)^dχd(h)(−m)= #(compatible pairs ofm-colorings and acyclic headings ofh).

In particular, (−1)^dχ_d(h)(−1) = #(acyclic headings of h).

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Outline

4 Reciprocity

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Hypergraphic polytope

|I|=d,R^d ∼=R^I,

for every noden_i pick a standard basis vectorb_i.

Forh= (I,E) define thehypergraphic polytope P(h)⊂R^d as:

P(h) =X

e∈E

∆e⊂R^d

where ∆_e = conv{b_i : n_i ∈e} for hyperedgese ⊆I.

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Vertex description of hypergraphic polytopes

Proposition ^(?)

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Vertex description of hypergraphic polytopes

Proposition ^(?)

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Vertex description of hypergraphic polytopes

Proof:

P(h) =X

e∈E

conv{b_i:n_i ∈e}= conv (

X

e∈E

{b_i:n_i ∈e} )

So,P(h) is convex hull of indegree vectors of headings. Left to show: indegree vector is vertex iff heading is acyclic.

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Vertex description of hypergraphic polytopes

Proof:

P(h) =X

e∈E

X

e∈E

{b_i:n_i ∈e} )

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Vertex description of hypergraphic polytopes

Proof:

P(h) =X

e∈E

X

e∈E

{b_i:n_i ∈e} )

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Vertex description of hypergraphic polytopes

Proof:

Claim 1: indegree vectors of cyclic headings can be written as convex combination.

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Vertex description of hypergraphic polytopes

Proof:

Claim 2: indegree vectors of acyclic headings cannot be written as convex combination.

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Subclasses

[AA17, AKT20]

• simplicial complexes ↔ simplicial complex polytopes

• graphs ↔graphical zonotopes

• building sets ↔nestohedra

• ...

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Outline

4 Reciprocity

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Normal fan

Fordirections y∈(R^d)^∗ and a polytopeP ⊂R^d call P_y :={x ∈ P : y(x)≥y(x⁰) for all x⁰ ∈ P}

they-maximal face. For a face F ofP define the normal cone NP(F) :={y ∈(R^d)^∗ : P_y ⊇F}.

Facts: • dimNP(F) =d−dimF = codimF

• F a face of G ⇔ NP(G) a face of NP(F)

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Standard permutahedra

Thestandard permutahedronπd is the convex hull ofd! vertices, namely, all the permutations of the point (1, . . . ,d).

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Braid fan and standard permutahedron

Thebraid arrangement B_d is the set of hyperplanes

H_i,j :={x∈(R^d)^∗:x_i =x_j}. Thebraid fan is the fan induced byB_d.

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Generalized Permutahedra

A polytopeP ⊂R^d is a generalized permutahedronif its normal fanN(P) is a coarsening of the braid fan.

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Submodular functions

A functionz: 2^I →Ris submodularif

z(S) +z(T)≥z(s∩T) +z(s∪T) for allS,T ⊆I.

Theorem

A polytopeP is a generalized permutahedron if and only if there exists a unique submodular functionz: 2^I →Rwith z(∅) = 0 such that

P = n

x ∈RI : X

i∈I

xi =z(I) and X

i∈T

xi ≤z(T) for T ⊆I o

.

Proof:

z(A) :=|{e∈E:e ∩A6=∅}|.

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Submodular functions

A functionz: 2^I →Ris submodularif

z(S) +z(T)≥z(s∩T) +z(s∪T) for allS,T ⊆I.

Theorem

A polytopeP is a generalized permutahedron if and only if there exists a unique submodular functionz: 2^I →Rwith z(∅) = 0 such that

P = n

x ∈RI : X

i∈I

xi =z(I) and X

i∈T

xi ≤z(T) for T ⊆I o

.

Corollary: Hypergraphic polytopes are generalized permutahedra.

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Relational submodular functions

A relationR ⊆I ×J defines a funtion zR: 2^I →Z≥0 by z_R(A) =|{b∈J: (a,b)∈R}| forA⊂I calledrelational submodular function.

A submodular functionz is relational

iff the associated generalized permutahedron is a hypergrphical polytope

iff z(∅) = 0 and for allA⊆I we havez(A)∈Z and P

K⊇A(−1)^|A−K|z(K)≤0.

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Relational submodular functions

A relationR ⊆I ×J defines a funtion zR: 2^I →Z≥0 by z_R(A) =|{b∈J: (a,b)∈R}| forA⊂I calledrelational submodular function.

Theorem ([AA17, Prop.19.4]) A submodular functionz is relational

iff the associated generalized permutahedron is a hypergrphical polytope

iff z(∅) = 0 and for allA⊆I we havez(A)∈Z and P

K⊇A(−1)^|A−K|z(K)≤0.

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Outline

4 Reciprocity

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Reciprocity for generalized permutahedra

χ_d(P)(m) := #

P-generic directionsy ∈(R^d)^∗ with y ∈[m]^d agrees with a polynomial inm of degreed.

Moreover, (−1)^dχ_d(P)(−m) = X

y∈[m]^d

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Reciprocity for generalized permutahedra

χ_d(P)(m) := #

(−1)^dχ_d(P)(−m) = X

y∈[m]^d

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Reciprocity for hypergraphs I

For a hypergraphh = (I,E) with I ={n₁, . . . ,n_d},m∈Z>0

χ_d(h)(m) := #(proper colorings of h with m colors) agrees with a polynomial inm of degreed.

Proof idea:

χ_d(h)(m) = # P(h)-generic directionsy ∈(R^d)^∗ withy ∈[m]^d

=χd(P(h)) (m)

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Reciprocity for hypergraphs I

χ_d(h)(m) := #(proper colorings of h with m colors) agrees with a polynomial inm of degreed.

Proof idea:

χd(h)(m) = # P(h)-generic directionsy ∈(R^d)^∗ withy ∈[m]^d

=χd(P(h)) (m)

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Reciprocity for hypergraphs I

Claim:

#(proper colorings of hwith m colors)

= # P(h)-generic directionsy∈(R^d)^∗withy ∈[m]^d .

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Reciprocity for hypergraphs II

(−1)^dχ_d(h)(−m)= #(compatible pairs ofm-colorings and acyclic headings ofh).

(−1)^dχ_d(h)(−m) = (−1)^dχ_d(P(h))(−m) = X

y∈[m]^d

# (vertices of P(h)y).

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Reciprocity for hypergraphs II

Recall:

(−1)^dχ_d(h)(−m) = (−1)^dχ_d(P(h))(−m) = X

y∈[m]^d

# (vertices of P(h)y).

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Reciprocity for hypergraphs II

Claim:

X

y∈[m]^d

# (vertices of P(h)y) = X

m-colorings

# (compatible acyclic headings of h)

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Reciprocity for hypergraphs (summary)

χ_d(h)(m) := #(proper colorings of h with m colors) agrees with a polynomial inm of degreed. Moreover,

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References I

Marcelo Aguiar and Federico Ardila,Hopf monoids and generalized permutahedra, arXiv:1709.07504.

Geir Agnarsson and Magn´us M. Halld´orsson,Strong colorings of hypergraphs, Approximation and Online Algorithms (Berlin, Heidelberg) (Giuseppe Persiano and Roberto Solis-Oba, eds.), Springer, 2005, pp. 253–266.

Jean-Christophe Aval, Th´eo Karaboghossian, and Adrian Tanasa,The Hopf monoid of hypergraphs and its sub-monoids: Basic invariant and reciprocity theorem, Electron. J. Combin. (2020), article P1.34, pp23.

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References II

Felix Breuer, Aaron Dall, and Martina Kubitzke,Hypergraph coloring complexes, Discrete Math. 312(2012), no. 16, 2407–2420.

Louis J. Billera, Ning Jia, and Victor Reiner,A quasisymmetric function for matroids, Eur. J. Combin.30(2009), no. 8, 1727–1757.

Csilla Bujt´as, Zsolt Tuza, and Vitaly Voloshin, Hypergraph colouring, Topics in Chromatic Graph Theory (Lowell W.

Beineke and Robin J. Wilson, eds.), Cambridge University Press, Cambridge, 2015, pp. 230–254.

Jean Cardinal and Stefan Felsner,Notes on Hypergraphic

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References III

Sophie Rehberg,Combinatorial reciprocity theorems for generalized permutahedra, hypergraphs, and pruned inside-out polytopes, arXiv: 2103.09073.

Lucas J. Rusnak,Oriented hypergraphs: Introduction and balance, Electron. J. Combin.20(2013), no. 3, article P48, 29 pp.

Thank you!

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Big Picture

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Polyhedral fans

For a complete fanN inR^d define thecodimension-one fanN^{co 1} N^{co 1} :={N∈ N : codimN≥1}

={N∈ N : dimN≤d −1}. For a normal fanN(P) we get

N^{co 1}(P) ={N_P(F) : F a face of P with dim(F)≥1}.

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Pruned inside-out polytopes

For a polytopeQ ⊂R^d and a complete fanN in R^d we call Q \[

N^{co 1}

= ]

N∈N, N full-dimensional

(Q ∩N^◦)

apruned inside-out polytopeand we call the connected components inQ \ S

N^{co 1}

regions.

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Pruned inside-out counting

Fort ∈Z>0 define the inner pruned Ehrhart function as

in_Q,N^{co 1}(t) :=#

t·

Q \[ N^{co 1}

∩Z^d .

in_Q,Nco 1 in_Q◦,N^{co 1}

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Pruned inside-out counting

Define thecumulative pruned Ehrhart functionfor t∈Z>0 as cu_Q,N^{co 1}(t) := X

y∈tQ∩Z^d

# (N∈ N,N full.dim., y ∈N) .

Note:

cu_Q,N^{co 1}(t) =

k

X

i=1

ehr_R

i(t)

and it is a polynomial if regionsRi are is integral.

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Pruned inside-out reciprocity

Theorem (R.)

For a polytopeQ ⊂R^d and a complete fanN in R^d we have (−1)^dim^Qin_Q^◦_,N^{co 1}(−t) = cu_Q,Nco 1(t).

Proof.

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Reciprocity for generalized permutahedra

χ_d(P)(m) := #

P-generic directionsy ∈(R^d)^∗ with y ∈[m]^d agrees with a polynomial inm of degreed.

Moreover, (−1)^dχ_d(P)(−m) = X

y∈[m]^d

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Reciprocity for generalized permutahedra

χ_d(P)(m) := #

(−1)^dχ_d(P)(−m) = X

y∈[m]^d

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Reciprocity for generalized permutahedra

Thm: polynomiality of

χ_d(P)(m) := # P-generic directions y ∈(R^d)^∗ with y ∈[m]^d .

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Reciprocity for generalized permutahedra

Thm: (−1)^dχd(P)(−m) = X

y∈[m]^d