The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck, San Francisco State University Thomas Zaslavsky, Binghamton University (SUNY) math.sfsu.edu/beck arXiv:math.CO/0309331
Flows on Graphs A : abelian group A -flow on a (bridgeless) graph Γ = ( V, E ) : mapping x : E → A such that for every node v ∈ V � � x ( e ) = x ( e ) h ( e )= v t ( e )= v h ( e ) := head of the edge e in a (fixed) orientation of Γ t ( e ) := tail The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 2
Flows on Graphs A : abelian group A -flow on a (bridgeless) graph Γ = ( V, E ) : mapping x : E → A such that for every node v ∈ V � � x ( e ) = x ( e ) h ( e )= v t ( e )= v h ( e ) := head of the edge e in a (fixed) orientation of Γ t ( e ) := tail k -flow : Z -flow with values in {− k + 1 , − k + 2 , . . . , k − 1 } The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 2
Flows on Graphs A : abelian group A -flow on a (bridgeless) graph Γ = ( V, E ) : mapping x : E → A such that for every node v ∈ V � � x ( e ) = x ( e ) h ( e )= v t ( e )= v h ( e ) := head of the edge e in a (fixed) orientation of Γ t ( e ) := tail k -flow : Z -flow with values in {− k + 1 , − k + 2 , . . . , k − 1 } ϕ 0 Γ ( k ) := # ( k -flows on Γ) ϕ Γ ( k ) := # ( nowhere-zero k -flows on Γ) ϕ Γ ( | A | ) := # ( nowhere-zero A -flows on Γ) The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 2
Flow Polynomials ϕ 0 Γ ( k ) := # ( k -flows on Γ) ϕ Γ ( k ) := # ( nowhere-zero k -flows on Γ) ϕ Γ ( | A | ) := # ( nowhere-zero A -flows on Γ) Theorem (Tutte 1954) ϕ Γ ( | A | ) is a polynomial in | A | . (Folklore) ϕ 0 Γ ( k ) is a polynomial in k . (Kochol 2002) ϕ Γ ( k ) is a polynomial in k . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 3
Flow Polynomials ϕ 0 Γ ( k ) := # ( k -flows on Γ) ϕ Γ ( k ) := # ( nowhere-zero k -flows on Γ) ϕ Γ ( | A | ) := # ( nowhere-zero A -flows on Γ) Theorem (Tutte 1954) ϕ Γ ( | A | ) is a polynomial in | A | . (Folklore) ϕ 0 Γ ( k ) is a polynomial in k . (Kochol 2002) ϕ Γ ( k ) is a polynomial in k . Remarks: ϕ Γ ( k ) > 0 if and only if ϕ Γ ( k ) > 0 . For a plane graph, ϕ Γ ( k ) = χ Γ ∗ ( k ) := # ( proper k -colorings of Γ ∗ ) . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 3
Enter Geometry Let C ⊆ R E denote the real cycle space of Γ defined by the equations � � x ( e ) = x ( e ) , h ( e )= v t ( e )= v and let ✷ = ( − 1 , 1) E . Then a k -flow on Γ is a point in ✷ ∩ C ∩ 1 k Z E , that is, a k -fractional lattice point in the polytope ✷ ∩ C . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 4
Enter Geometry Let C ⊆ R E denote the real cycle space of Γ defined by the equations � � x ( e ) = x ( e ) , h ( e )= v t ( e )= v and let ✷ = ( − 1 , 1) E . Then a k -flow on Γ is a point in ✷ ∩ C ∩ 1 k Z E , that is, a k -fractional lattice point in the polytope ✷ ∩ C . Moreover, if we let H denote the set of coordinate hyperplanes in R E , then a nowhere-zero k -flow on Γ is a k -fractional lattice point in � ( ✷ ∩ C ) \ H , an instance of an inside-out polytope. The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 4
A Simple Example 2 K 2 The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 5
A Simple Example y x x y 2 K 2 The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 5
A Simple Example y x x y 2 K 2 φ 0 2 K 2 ( k ) = 2 k − 1 The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 5
A Simple Example y x x y 2 K 2 φ 2 K 2 ( k ) = 2 k − 2 The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 5
Ehrhart Polynomials P ⊂ R d – convex integral polytope P ∩ 1 � t Z d � For t ∈ Z > 0 let Ehr P ( t ) := # The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 6
Ehrhart Polynomials P ⊂ R d – convex integral polytope P ∩ 1 � t Z d � For t ∈ Z > 0 let Ehr P ( t ) := # Theorem (Ehrhart 1962) Ehr P ( t ) is a polynomial in t of degree dim P with leading term vol P (normalized to aff P ∩ Z d ). (Macdonald 1971) ( − 1) dim P Ehr P ( − t ) enumerates the interior lattice points P ◦ ∩ 1 t Z d . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 6
Ehrhart Polynomials P ⊂ R d – convex integral polytope P ∩ 1 � t Z d � For t ∈ Z > 0 let Ehr P ( t ) := # Theorem (Ehrhart 1962) Ehr P ( t ) is a polynomial in t of degree dim P with leading term vol P (normalized to aff P ∩ Z d ). (Macdonald 1971) ( − 1) dim P Ehr P ( − t ) enumerates the interior lattice points P ◦ ∩ 1 t Z d . Ehrhart Theory has recently seen a flurry of applications in various areas of mathematics. One class of applications comes from the enumeration of lattice points in a polytope P but off a hyperplane arrangement H —an inside-out polytope ( P , H ) . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 6
Polynomiality Theorem (Folklore) ϕ 0 Γ ( k ) is a polynomial in k . (Kochol 2002) ϕ Γ ( k ) is a polynomial in k . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 7
Polynomiality Theorem (Folklore) ϕ 0 Γ ( k ) is a polynomial in k . (Kochol 2002) ϕ Γ ( k ) is a polynomial in k . Proof: Recall that C ⊆ R E is defined by � h ( e )= v x ( e ) = � t ( e )= v x ( e ) , ✷ = ( − 1 , 1) E , and H contains the coordinate hyperplanes in R E ; then � � ✷ ∩ C ∩ 1 ϕ 0 k Z E Γ ( k ) = # �� ∩ 1 � � � k Z E ϕ Γ ( k ) = # ✷ ∩ C \ H . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 7
Polynomiality Theorem (Folklore) ϕ 0 Γ ( k ) is a polynomial in k . (Kochol 2002) ϕ Γ ( k ) is a polynomial in k . Proof: Recall that C ⊆ R E is defined by � h ( e )= v x ( e ) = � t ( e )= v x ( e ) , ✷ = ( − 1 , 1) E , and H contains the coordinate hyperplanes in R E ; then � ✷ ∩ C ∩ 1 � ϕ 0 k Z E Γ ( k ) = # �� � ∩ 1 � � k Z E ϕ Γ ( k ) = # ✷ ∩ C \ H . The matrix defining the cycle space C is totally unimodular, and hence ✷ ∩ C is an integral polytope. For the same reason, any of the connected components of ✷ ∩ C \ � H is an integral polytope. The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 7
Totally Cyclic Orientations An orientation of Γ is totally cyclic if every edge lies in a coherent circle, that is, where the edges are oriented in a consistent direction around the circle. A totally cyclic orientation τ and a flow x are compatible if x ≥ 0 when it is expressed in terms of τ . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 8
Totally Cyclic Orientations An orientation of Γ is totally cyclic if every edge lies in a coherent circle, that is, where the edges are oriented in a consistent direction around the circle. A totally cyclic orientation τ and a flow x are compatible if x ≥ 0 when it is expressed in terms of τ . Recall ϕ Γ ( k ) := # ( nowhere-zero k -flows ) Theorem (B–Z) | ϕ Γ ( − k ) | equals the number of pairs ( τ, x ) consisting of a totally cyclic orientation τ and a compatible ( k + 1) -flow x . In particular, | ϕ Γ (0) | counts the totally cyclic orientations of Γ . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 8
Totally Cyclic Orientations An orientation of Γ is totally cyclic if every edge lies in a coherent circle, that is, where the edges are oriented in a consistent direction around the circle. A totally cyclic orientation τ and a flow x are compatible if x ≥ 0 when it is expressed in terms of τ . Recall ϕ Γ ( k ) := # ( nowhere-zero k -flows ) Theorem (B–Z) | ϕ Γ ( − k ) | equals the number of pairs ( τ, x ) consisting of a totally cyclic orientation τ and a compatible ( k + 1) -flow x . In particular, | ϕ Γ (0) | counts the totally cyclic orientations of Γ . φ 2 K 2 ( k ) = 2 k − 2 The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 8
A (Classic) Dual Theorem Γ = ( V, E ) – (loopless) graph k -coloring of Γ : mapping x : V → { 1 , 2 , . . . , k } Proper k -coloring of Γ : k -coloring such that x i � = x j if there is an edge ij The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 9
A (Classic) Dual Theorem Γ = ( V, E ) – (loopless) graph k -coloring of Γ : mapping x : V → { 1 , 2 , . . . , k } Proper k -coloring of Γ : k -coloring such that x i � = x j if there is an edge ij Theorem (Birkhoff 1912, Whitney 1932) χ Γ ( k ) := # ( proper k -colorings of Γ) is a polynomial in k . The Enumeration of Nowhere-Zero Integral Flows on Graphs Matthias Beck 9
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