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The Tutte polynomial and a bijection between subgraphs and orientations Olivier Bernardi - CRM Combinatorics Seminar at CRM, October 2006, Barcelona Content of the talk The Tutte Polynomial and its specializations A new characterization of the

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The Tutte polynomial and a bijection between subgraphs and orientations

Olivier Bernardi - CRM

Combinatorics Seminar at CRM, October 2006, Barcelona

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Content of the talk

The Tutte Polynomial and its specializations A new characterization of the Tutte Polynomial (based on the embeddings of graphs) A nice bijection between subgraphs and orientations

CRM, October 2006 Olivier Bernardi - CRM – p.1/31

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The Tutte polynomial

[Whitney ∼30 - Tutte ∼40]

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The Tutte polynomial

Graph G = (V, E).

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The Tutte polynomial

Graph G = (V, E). Spanning subgraph: contains every vertex and a subset of the edges.

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The Tutte polynomial

Graph G = (V, E). Spanning subgraph: contains every vertex and a subset of the edges. |S| = 5 c(S) = 3 We denote by:

  • |S| the number of edges,
  • c(S) the number of connected components.

CRM, October 2006 Olivier Bernardi - CRM – p.3/31

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The Tutte polynomial

Definition: The Tutte polynomial of the graph G = (V, E) is TG(x, y) =

  • S⊆G

(x − 1)c(S)−c(G)(y − 1)|S|+c(S)−|V |.

CRM, October 2006 ▽Olivier Bernardi - CRM – p.4/31

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The Tutte polynomial

Definition: The Tutte polynomial of the graph G = (V, E) is TG(x, y) =

  • S⊆G

(x − 1)c(S)−c(G)(y − 1)|S|+c(S)−|V |. Example : TK3(x, y) = x2 + x + y (x − 1)2 (y − 1) + 3 · (x − 1) + TK3(x, y) = 3 · 1 +

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The Tutte polynomial

Definition: The Tutte polynomial of the graph G = (V, E) is TG(x, y) =

  • S⊆G

(x − 1)c(S)−c(G)(y − 1)|S|+c(S)−|V |.

  • c(S) − c(G): renormalized number of connected

components.

  • |S| + c(S) − |V |: cyclomatic number.

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The Tutte polynomial of the union of two disjoint graphs : TG1∪G2(x, y) = TG1(x, y) × TG2(x, y).

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The Tutte polynomial of the union of two disjoint graphs : TG1∪G2(x, y) = TG1(x, y) × TG2(x, y). = ⇒ We restrict our attention to connected graphs.

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Relations of induction

Proposition [Tutte 1947]:

  • T•(x, y) = 1,
  • TG(x, y) = yTG\e(x, y) if e is a loop,
  • TG(x, y) = xTG/e(x, y) if e is a ismuth,
  • TG(x, y) = TG\e(x, y) + TG/e(x, y) otherwise.

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Specializations

TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |.

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Specializations

TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |.

  • TG(2, 2) = 2|E| = Number of sugraphs.

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Specializations

TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |.

  • TG(2, 2) = 2|E| = Number of sugraphs.
  • TG(1, 2) = Number of connected subgraphs.

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Specializations

TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |.

  • TG(2, 2) = 2|E| = Number of sugraphs.
  • TG(1, 2) = Number of connected subgraphs.
  • TG(2, 1) = Number of forests.

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Specializations

TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |.

  • TG(2, 2) = 2|E| = Number of sugraphs.
  • TG(1, 2) = Number of connected subgraphs.
  • TG(2, 1) = Number of forests.
  • TG(1, 1) = Number of spanning trees.

CRM, October 2006 ▽Olivier Bernardi - CRM – p.7/31

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Specializations

TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |.

  • TG(2, 2) = 2|E| = Number of sugraphs.
  • TG(1, 2) = Number of connected subgraphs.
  • TG(2, 1) = Number of forests.
  • TG(1, 1) = Number of spanning trees.

Other specializations: chromatic polynomial, flow polynomial, acyclic orientations, strongly connected

  • rientations, bipolar orientations, score vectors, sandpile

configurations, Potts model, reliability polynomial. . .

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Specializations: Example

TG(x, y) = x2 + x + y TG(2, 2) = 8 = #subgraphs = #orientations.

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Specializations: Example

TG(x, y) = x2 + x + y TG(1, 2) = 4 = #connected subgraphs

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Specializations: Example

TG(x, y) = x2 + x + y TG(1, 2) = 4 = #connected subgraphs = #root-connected orientations.

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Specializations: Example

TG(x, y) = x2 + x + y TG(2, 1) = 7 = #forests

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Specializations: Example

TG(x, y) = x2 + x + y 0 0 1 1 1 1 1 1 1 2 2 1 2 2 1 1 1 1 2 2 TG(2, 1) = 7 = #forests = #score vectors.

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A new characterization

  • f the Tutte polynomial

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Several characterizations

Tutte polynomial: TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |. Subgraph expansion [Tutte, Whitney].

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Several characterizations

Tutte polynomial: TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |. Subgraph expansion [Tutte, Whitney]. Induction definition [Tutte 47] Spanning tree expansion [Tutte 54] Orientation expansion [Las Vergnas 84]

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Several characterizations

Tutte polynomial: TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |. Subgraph expansion [Tutte, Whitney]. Induction definition [Tutte 47] Spanning tree expansion [Tutte 54] Orientation expansion [Las Vergnas 84] + [Gessel & Wang 79] [Gessel & Sagan 96] [Kostic & Yan 06]

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Spanning trees

A spanning tree T.

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Spanning trees

Fundamental cycle of an external edge e.

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Spanning trees

Fundamental cycle of an external edge e.

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Spanning trees

Fundamental cocycle of an internal edge e.

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Spanning trees

Fundamental cocycle of an internal edge e.

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Embeddings

A combinatorial embedding of a graph is a choice of a cyclic ordering of the half-edges around each vertex.

CRM, October 2006 ▽Olivier Bernardi - CRM – p.12/31

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Embeddings

A combinatorial embedding of a graph is a choice of a cyclic ordering of the half-edges around each vertex. An embedding is rooted if an half-edge is distinguished as the root.

CRM, October 2006 Olivier Bernardi - CRM – p.12/31

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Embeddings

Theorem [Mohar & Thomasen]: There is a one-to-one correspondence between combinatorial embeddings and topological embeddings in (orientable) surfaces considered up to homeomorphism. ⇐ ⇒

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Embeddings

Theorem [Mohar & Thomasen]: There is a one-to-one correspondence between combinatorial embeddings and topological embeddings in (orientable) surfaces considered up to homeomorphism. ⇐ ⇒

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Moving in an embedded graph

To move in an embedded graph, one can

  • Turn around a vertex:
  • Cross an edge :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn.

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

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The tour of a spanning tree

Start from the root. Iterate the move : External half-edge = ⇒ Turn. Internal half-edge = ⇒ Cross + turn. Example :

CRM, October 2006 Olivier Bernardi - CRM – p.15/31

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Tour: graphical interpretation

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Activities of spanning trees

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Activities of spanning trees

Consider the order of appearance of the edges around the tree.

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Activities of spanning trees

5 4 6 2 1 3 Consider the order of appearance of the edges around the tree.

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Activities of spanning trees

5 4 6 2 1 3 Consider the order of appearance of the edges around the tree. An internal/external edge is active if it is minimal in its fundamental cocycle/cycle.

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Activities of spanning trees

5 4 6 2 1 3 Consider the order of appearance of the edges around the tree. We denote by:

  • i(A) the number of internal active edges,
  • e(A) the number of external active edges.

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Tutte polynomial

Theorem [B.] : The Tutte polynomial is equal to : TG(x, y) =

  • T spanning tree

xi(T)ye(T).

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Tutte polynomial

Theorem [B.] : The Tutte polynomial is equal to : TG(x, y) =

  • T spanning tree

xi(T)ye(T). Example : TK3(x, y) = x2 + x + y. TK3(x, y) = x2 + x + y

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Tutte polynomial

Theorem [B.] : The Tutte polynomial is equal to : TG(x, y) =

  • T spanning tree

xi(T)ye(T). Subgraph expansion [Brylawsky] : TG(x, y) =

  • S⊆G

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V |.

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From spanning trees to subgraphs

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From spanning trees to subgraphs

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From spanning trees to subgraphs

T T − T +

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Partition of the subgraphs

Proposition [B.] : 2E =

  • T spanning tree

[T −; T +]

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Partition of the subgraphs

Proposition [B.] : 2E =

  • T spanning tree

[T −; T +] Example :

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Partition of the subgraphs

Proposition [B.] : 2E =

  • T spanning tree

[T −; T +] Example :

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Partition of the subgraphs

Proposition [B.] : 2E =

  • T spanning tree

[T −; T +] Proposition [B.] :

  • S∈[T −,T +]

(x − 1)c(S)−1(y − 1)|S|+c(S)−|V | = xi(T)ye(T)

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Bijection subgraphs ⇐ ⇒ orientations

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Φ : Subgraphs → Orientations

For spanning trees:

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Φ : Subgraphs → Orientations

For spanning trees: Internal edges are oriented from father to son.

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Φ : Subgraphs → Orientations

For spanning trees: Internal edges are oriented from father to son. External edges are oriented in such a way their heads appear before their tails around the tree.

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Φ : Subgraphs → Orientations

For spanning trees: Internal edges are oriented from father to son. External edges are oriented in such a way their heads appear before their tails around the tree.

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Activity et acyclicity

Remark : The fundamental cycle of an external active edge is oriented.

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Activity et acyclicity

Remark : The fundamental cycle of an external active edge is oriented. Hence, Φ(T) acyclic = ⇒ T internal.

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Activity et acyclicity

Remark : The fundamental cycle of an external active edge is oriented. Hence, Φ(T) acyclic ⇐ ⇒ T internal.

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Activity et strong connectivity

Remark : The fundamental cocycle of an internal active edge is oriented.

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Activity et strong connectivity

Remark : The fundamental cocycle of an internal active edge is oriented. Hence, Φ(T) strongly connected = ⇒ T external.

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Activity et strong connectivity

Remark : The fundamental cocycle of an internal active edge is oriented. Hence, Φ(T) strongly connected ⇐ ⇒ T external.

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Φ : Subgraphs → Orientations

For any subgraph S:

Φ ?

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Φ : Subgraphs → Orientations

For any subgraph S: The subgraph S is in an interval [T −, T +].

Φ ? Φ

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Φ : Subgraphs → Orientations

For any subgraph S: The subgraph S is in an interval [T −, T +]. Reverse the fundamental cycle/cocycle of the edges in T△S.

Φ Φ

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Φ : Subgraphs → Orientations

Theorem [B.] : For any graph G, the mapping Φ establishes a bijection between subgraphs and

  • rientations.

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Φ : Subgraphs → Orientations

Theorem [B.] : For any graph G, the mapping Φ establishes a bijection between subgraphs and

  • rientations.

Example :

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Φ : Subgraphs → Orientations

Theorem [B.] : For any graph G, the mapping Φ establishes a bijection between subgraphs and

  • rientations.

Example :

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Specializations

Subgraphs Orientations general connectedexternal general root- connected strongly connected mini general forest internal acyclic mal general

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Specializations

Connected: TG(1, 2)

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Specializations

Connected: TG(1, 2) :Root-connected.

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Specializations

External: TG(0, 2)

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Specializations

External: TG(0, 2) :Strongly-connected.

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Specializations

Forests: TG(2, 1)

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Specializations

Forests: TG(2, 1) :Minimal (⇐ ⇒ score vectors).

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Specializations

Internal: TG(2, 0)

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Specializations

Internal: TG(2, 0) :Acyclic.

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Specializations

Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

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Specializations

Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

  • Connected subgraphs and root-connected orientations.

TG(1, 2) [Gioan 06]

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Specializations

Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

  • Connected subgraphs and root-connected orientations.

TG(1, 2) [Gioan 06]

  • Forests and minimal orientations (⇐

⇒ score vectors). TG(2, 1) [Stanley 80]

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Specializations

Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

  • Connected subgraphs and root-connected orientations.

TG(1, 2) [Gioan 06]

  • Forests and minimal orientations (⇐

⇒ score vectors). TG(2, 1) [Stanley 80]

  • Trees and minimal root-connected orientations

(⇐ ⇒ root-connected score vectors). TG(1, 1)[Gioan 06]

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Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

  • Internal subgraphs and acyclic orientations.

TG(2, 0) [Winder 66, Stanley 73, Gessel & Sagan 96]

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Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

  • Internal subgraphs and acyclic orientations.

TG(2, 0) [Winder 66, Stanley 73, Gessel & Sagan 96]

  • External subgraphs and strongly-connected
  • rientations.

TG(0, 2) [Las Vergnas 84, Gioan & Las Vergnas 06]

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Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

  • Internal subgraphs and acyclic orientations.

TG(2, 0) [Winder 66, Stanley 73, Gessel & Sagan 96]

  • External subgraphs and strongly-connected
  • rientations.

TG(0, 2) [Las Vergnas 84, Gioan & Las Vergnas 06]

  • Internal trees and acyclic root-connected orientations.

TG(1, 0) [Greene & Zaslavsky 83, Gessel & Sagan 96, Gebhard & Sagan 00]

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Theorem [B.] : For any graph G, the mapping Φ induces a bijection between:

  • Internal subgraphs and acyclic orientations.

TG(2, 0) [Winder 66, Stanley 73, Gessel & Sagan 96]

  • External subgraphs and strongly-connected
  • rientations.

TG(0, 2) [Las Vergnas 84, Gioan & Las Vergnas 06]

  • Internal trees and acyclic root-connected orientations.

TG(1, 0) [Greene & Zaslavsky 83, Gessel & Sagan 96, Gebhard & Sagan 00]

  • External trees and minimal strongly-connected
  • rientations (⇐

⇒ strongly-connected score vectors). TG(0, 1) [Gioan 06]

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Thanks.

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