Polynomial graph and matroid invariants from graph homomorphisms - PowerPoint PPT Presentation

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants From B -tensions and B -flows to a B -Tutte? Beyond polynomials? Open problems Polynomial graph and matroid invariants from graph homomorphisms Delia Garijo 1 Andrew Goodall 2 Patrice Ossona de Mendez 3 ril 2 Guus Regts 4 Jarik Neˇ setˇ ıs Vena 2 and Llu´ 1 University of Seville 2 Charles University, Prague 3 CAMS, CNRS/EHESS, Paris 4 University of Amsterdam 14 June 2016 Schloss Dagstuhl

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Chromatic polynomial Definition (Evaluation at positive integers) k ∈ N , P ( G ; k ) = # { proper vertex k -colourings of G } .

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Chromatic polynomial Definition (Evaluation at positive integers) k ∈ N , P ( G ; k ) = # { proper vertex k -colourings of G } . e ∈ E ( G ) : P ( G ; k ) = P ( G \ e ; k ) − P ( G / e ; k )

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Flow polynomial Definition (Evaluation at positive integers) k ∈ N , F ( G ; k ) = # { nowhere-zero Z k -flows of G } .  F ( G / e ) − F ( G \ e ) e ordinary   e ∈ E ( G ) : F ( G ; k ) = 0 e a bridge   ( k − 1) F ( G \ e ) e a loop

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Tutte polynomial Definition (Subgraph expansion) For graph G = ( V , E ), � ( x − 1) r ( E ) − r ( A ) ( y − 1) | A |− r ( A ) , T ( G ; x , y ) = A ⊆ E where r ( A ) is the rank of the spanning subgraph ( V , A ) of G .

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Tutte polynomial Definition (Subgraph expansion) For graph G = ( V , E ), � ( x − 1) r ( E ) − r ( A ) ( y − 1) | A |− r ( A ) , T ( G ; x , y ) = A ⊆ E where r ( A ) is the rank of the spanning subgraph ( V , A ) of G .   T ( G / e ; x , y ) + T ( G \ e ; x , y ) e ordinary  T ( G ; x , y ) = xT ( G / e ; x , y ) e a bridge   yT ( G \ e ; x , y ) e a loop , and T ( G ; x , y ) = 1 if G has no edges.

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Definition Graphs G , H . f : V ( G ) → V ( H ) is a homomorphism from G to H if uv ∈ E ( G ) ⇒ f ( u ) f ( v ) ∈ E ( H ).

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Definition Graphs G , H . f : V ( G ) → V ( H ) is a homomorphism from G to H if uv ∈ E ( G ) ⇒ f ( u ) f ( v ) ∈ E ( H ). Definition H with adjacency matrix ( a s , t ), weight a s , t on st ∈ E ( H ), � � hom ( G , H ) = a f ( u ) , f ( v ) . f : V ( G ) → V ( H ) uv ∈ E ( G )

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Graph polynomials From B -tensions and B -flows to a B -Tutte? Graph homomorphisms Beyond polynomials? Open problems Definition Graphs G , H . f : V ( G ) → V ( H ) is a homomorphism from G to H if uv ∈ E ( G ) ⇒ f ( u ) f ( v ) ∈ E ( H ). Definition H with adjacency matrix ( a s , t ), weight a s , t on st ∈ E ( H ), � � hom ( G , H ) = a f ( u ) , f ( v ) . f : V ( G ) → V ( H ) uv ∈ E ( G ) hom ( G , H ) = # { homomorphisms from G to H } = # { H -colourings of G } when H simple ( a s , t ∈ { 0 , 1 } ) or multigraph ( a s , t ∈ N )

bc bc b b b b bc bc b b b b b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 1 ( K k )

b bc b b b b bc bc b b b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 1 ( K k ) hom ( G , K k ) = P ( G ; k ) chromatic polynomial

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Problem Which sequences ( H k ) of graphs are such that, for all graphs G , for each k ∈ N we have hom ( G , H k ) = p ( G ; k ) for a fixed polynomial p ( G )?

Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Problem Which sequences ( H k ) of graphs are such that, for all graphs G , for each k ∈ N we have hom ( G , H k ) = p ( G ; k ) for a fixed polynomial p ( G )? Example For all graphs G , hom ( G , K k ) = P ( G ; k ) is the evaluation of the chromatic polynomial of G at k .

bc b b b b b bc bc b b b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 2: add loops ( K 1 k ) hom ( G , K 1 k ) = k | V ( G ) |

b bc b b b bc bc b b b b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 3: add ℓ loops ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ( K ℓ k ) � hom ( G , K ℓ ℓ # { uv ∈ E ( G ) | f ( u )= f ( v ) } k ) = f : V ( G ) → [ k ]

b bc b b b bc bc b b b b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 3: add ℓ loops ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ( K ℓ k ) � hom ( G , K ℓ ℓ # { uv ∈ E ( G ) | f ( u )= f ( v ) } k ) = f : V ( G ) → [ k ] = k c ( G ) ( ℓ − 1) r ( G ) T ( G ; ℓ − 1+ k ℓ − 1 , ℓ ) (Tutte polynomial)

b bc b b b bc bc b b b b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 4: add loops weight 1 − k − 3 − 2 − 1 − 3 − 2 − 1 − 2 − 3 − 3 ( K 1 − k ) k � hom ( G , K 1 − k (1 − k ) # { uv ∈ E ( G ) | f ( u )= f ( v ) } ) = k f : V ( G ) → [ k ]

b bc b b b bc bc b b b b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 4: add loops weight 1 − k − 3 − 2 − 1 − 3 − 2 − 1 − 2 − 3 − 3 ( K 1 − k ) k � hom ( G , K 1 − k (1 − k ) # { uv ∈ E ( G ) | f ( u )= f ( v ) } ) = k f : V ( G ) → [ k ] = ( − 1) | E ( G ) | k | V ( G ) | F ( G ; k ) (flow polynomial)

b b bc b b b b bc bc bc bc b b bc b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 5 ( K � k 2 ) = ( Q k ) (hypercubes)

b b bc b b b b bc bc bc bc b b bc b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 5 ( K � k 2 ) = ( Q k ) (hypercubes) Proposition (Garijo, G., Neˇ setˇ ril, 2015) hom ( G , Q k ) = p ( G ; k , 2 k ) for bivariate polynomial p ( G )

b bc b b b b bc bc b b b bc b Counting graph homomorphisms Sequences giving graph polynomials Cycle matroid invariants Examples From B -tensions and B -flows to a B -Tutte? Strongly polynomial sequences of graphs Beyond polynomials? Open problems Example 6 ( C k ) hom ( C 3 , C 3 ) = 6, hom ( C 3 , C k ) = 0 when k = 2 or k ≥ 4