[PPT] - CONSTRUCTION OF SNARKS WITH CIRCULAR FLOW NUMBER 5 Giuseppe PowerPoint Presentation

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

CONSTRUCTION OF SNARKS WITH CIRCULAR FLOW NUMBER 5

Giuseppe Mazzuoccolo University of Verona (Italy)

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Nowhere-zero Circular Flows

Let r ≥ 2 be a real number. A circular nowhere-zero r-flow (for short r-CNZF) in a graph G is

An assignment + An orientation f : E → [1, r − 1] D such that for every vertex v ∈ V ,

e∈E +(v)

f (e) =

e∈E −(v)

f (e) .

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

TRIVIAL Necessary Condition

Necessary condition If G has a r-CNZF, then G is BRIDGELESS.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

TRIVIAL Necessary Condition

Necessary condition If G has a r-CNZF, then G is BRIDGELESS.

BRIDGE

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Definition The CIRCULAR FLOW NUMBER φc(G) of a bridgeless graph G is the infimum of the set of numbers r, for which G admits an r-CNZF.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Definition The CIRCULAR FLOW NUMBER φc(G) of a bridgeless graph G is the infimum of the set of numbers r, for which G admits an r-CNZF. NOTE: It is known (Goddyn-Tarsi-Zhang) that φc(G) does exist & it is a minimum and a rational number.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

5-flow Conjecture (W.TUTTE - 1954)

Tutte’s Conjecture (1954) Every bridgeless graph has a 5-(Circular)NZF. Seymour’s Theorem (1981) Every bridgeless graph has a 6-(Circular)NZF.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

CIRCULAR NOWHERE-ZERO FLOWS IN CUBIC GRAPHS

CONJECTURE

5−FLOW NO EXAMPLE NO EXAMPLE SEYMOUR

THEOREM

BIPARTITE IFF 3−EDGE COLORABLE

NOT

3 4 5 6

IFF 3−EDGE COLORABLE

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

CIRCULAR NOWHERE-ZERO FLOWS IN CUBIC GRAPHS

CONJECTURE

5−FLOW NO EXAMPLE

3 4 5 6

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

CIRCULAR NOWHERE-ZERO FLOWS IN CUBIC GRAPHS

CONJECTURE

5−FLOW NO EXAMPLE PETERSEN GRAPH

3 4 5 6

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

CIRCULAR NOWHERE-ZERO FLOWS IN CUBIC GRAPHS

CONJECTURE

5−FLOW NO EXAMPLE

3 4 5 6

GRAPH PETERSEN

+ TRIVIAL EXAMPLES

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

CIRCULAR NOWHERE-ZERO FLOWS IN CUBIC GRAPHS

CONJECTURE

5−FLOW NO EXAMPLE

3 4 5 6

GRAPH PETERSEN

+ TRIVIAL EXAMPLES

G P Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

CIRCULAR NOWHERE-ZERO FLOWS IN CUBIC GRAPHS

CONJECTURE

5−FLOW NO EXAMPLE

3 4 5 6

GRAPH PETERSEN

+ TRIVIAL EXAMPLES

G P Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

✭✭✭✭✭✭✭✭✭✭ ✭

CUBIC GRAPHS SNARKS WITH CIRCULAR FLOW NUMBER 5?

SNARK =            CUBIC GRAPH CHROMATIC INDEX 4 GIRTH ≥ 5 CYCLICALLY 4-EDGE-CONNECTED

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SNARKS WITH CIRCULAR FLOW NUMBER 5?

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SNARKS WITH CIRCULAR FLOW NUMBER 5?

Mohar’s conjecture - 2003 Petersen graph is the unique snark with φc = 5.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SNARKS WITH CIRCULAR FLOW NUMBER 5?

Mohar’s conjecture - 2003 Petersen graph is the unique snark with φc = 5. Macajova,Raspaud - 2006 Infinite family of snarks with φc = 5.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SNARKS WITH CIRCULAR FLOW NUMBER 5?

Mohar’s conjecture - 2003 Petersen graph is the unique snark with φc = 5. Macajova,Raspaud - 2006 Infinite family of snarks with φc = 5. Esperet, G.M., Tarsi - 2015

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SNARKS WITH CIRCULAR FLOW NUMBER 5?

Mohar’s conjecture - 2003 Petersen graph is the unique snark with φc = 5. Macajova,Raspaud - 2006 Infinite family of snarks with φc = 5. Esperet, G.M., Tarsi - 2015 Larger family of counterexamples

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SNARKS WITH CIRCULAR FLOW NUMBER 5?

Mohar’s conjecture - 2003 Petersen graph is the unique snark with φc = 5. Macajova,Raspaud - 2006 Infinite family of snarks with φc = 5. Esperet, G.M., Tarsi - 2015 Larger family of counterexamples The corresponding recognition problem is NP-complete.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SNARKS WITH CIRCULAR FLOW NUMBER 5?

Mohar’s conjecture - 2003 Petersen graph is the unique cyclically 5-edge-connected snark with φc = 5. Macajova,Raspaud - 2006 Infinite family of snarks with φc = 5. Esperet, G.M., Tarsi - 2015 Larger family of counterexamples The corresponding recognition problem is NP-complete.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Modular r-flow

A circular nowhere-zero modular-r-flow (r-mcnzf) is an analogue of an r-cnzf, where the additive group of real numbers is replaced by the additive group of R/rZ.

Proposition The existence of a circular nowhere-zero r-flow in a graph G is equivalent to that of an r-mcnzf.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

OPEN k-CAPACITY, CPk(Gu,v)

Definition Let k be a positive integer. The open k-capacity CPk(Gu,v) of Gu,v is a subset of R/kZ, defined as follows: Add to G an additional edge e0 / ∈ E(G) with endvertices u and v, and set:

CPk(Gu,v) = {f (e0) | f is a mod k flow in G ∪ eo and f : E(G) → (1, k − 1)}

G G

u v u,v

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

OPEN k-CAPACITY, CPk(Gu,v)

Definition Let k be a positive integer. The open k-capacity CPk(Gu,v) of Gu,v is a subset of R/kZ, defined as follows: Add to G an additional edge e0 / ∈ E(G) with endvertices u and v, and set:

CPk(Gu,v) = {f (e0) | f is a mod k flow in G ∪ eo and f : E(G) → (1, k − 1)}

G G

u v u,v e0

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Example: determine CP5 of Petersen minus an edge

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Example: determine CP5 of Petersen minus an edge

(1,4) v u (?,?)

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Example: determine CP5 of Petersen minus an edge

(1,4) v u (?,?) ?

CP5(Gu,v) ⊆ (4,1)

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

PROPERTIES OF CPk(Gu,v)

CPk(Gu,v) is a symmetric subset of R/kZ. If t ∈ CPk(Gu,v), then −t ∈ CPk(Gu,v).

1 2 k−1

R/kZ t

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

PROPERTIES OF CPk(Gu,v)

CPk(Gu,v) is a symmetric subset of R/kZ. If t ∈ CPk(Gu,v), then −t ∈ CPk(Gu,v).

1 2 k−1

R/kZ −t t

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

PROPERTIES OF CPk(Gu,v)

CPk(Gu,v) is a symmetric subset of R/kZ. If t ∈ CPk(Gu,v), then −t ∈ CPk(Gu,v). If t ∈ CPk(Gu,v), then (a, b) ∈ CPk(Gu,v) (with a, b integers).

1 2 k−1 b a

R/kZ −t t

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

PROPERTIES OF CPk(Gu,v)

CPk(Gu,v) is a symmetric subset of R/kZ. If t ∈ CPk(Gu,v), then −t ∈ CPk(Gu,v). If t ∈ CPk(Gu,v), then (a, b) ∈ CPk(Gu,v) (with a, b integers).

1 2 k−1 b a

R/kZ −t t

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

PROPERTIES OF CPk(Gu,v)

CPk(Gu,v) is a symmetric subset of R/kZ. If t ∈ CPk(Gu,v), then −t ∈ CPk(Gu,v). If t ∈ CPk(Gu,v), then (a, b) ∈ CPk(Gu,v) (with a, b integers).

1 2 k−1 b a

R/kZ −t t

CPk(Gu,v) is symmetric and union of open intervals (with integer endpoints).

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SIk and GIk

Definition Let SIk denote the set of all unions of open intervals which form symmetric subsets of R/kZ. Some elements of SI5:

1 2 4 3

R/5Z (1,4)

1 2 4 3

R/5Z (4,1)

1 2 4 3

R/5Z (1,2)U(3,4)

1 2 4 3

R/5Z (0,0)

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SIk and GIk

Definition We say that a set A ∈ SIk is graphic, if there exists Gu,v with

pen k-capacity A. The set of all graphic members of SIk is

denoted by GIk.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

SIk and GIk

Definition We say that a set A ∈ SIk is graphic, if there exists Gu,v with

pen k-capacity A. The set of all graphic members of SIk is

denoted by GIk. k=5

1 2 4 3 1 2 4 3

(2,3)U(3,2) (4,1) u v

(4,1) is GRAPHIC WE HAVE NO EXAMPLE (2,3)U(3,2) GRAPHIC???

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Generation of GI5

Parallel join

G’ G’’

u,v

CP (G’ )=A

k

G’ G’’

u,v

CP (G’’ )=B A+B

k

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Generation of GI5

Serial join

u,v

CP (G’ )=A

k

u,v

CP (G’’ )=B

k

G’ G’’ G’ G’’ Α Β

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Generation of GI5

(1, 4) (4, 1) (3, 2) (0, 0) (the Petersen graph) ∅ R/5Z (1, 2) ∪ (3, 4) (4, 0) ∪ (0, 1) (3, 0) ∪ (0, 2) (1, 4) (the thick edge) (4, 1) ∪ (2, 3) (2, 3) (4, 0) ∪ (0, 1) ∪ (2, 3) Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Construction of graphs with Φc ≥ 5

Definition The measure Me(A), A ∈ GI5, is the number of unit intervals contained in A.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Construction of graphs with Φc ≥ 5

Definition The measure Me(A), A ∈ GI5, is the number of unit intervals contained in A. Me((1, 2) ∪ (3, 4)) = 2 Me((1, 4)) = 3 Me((4, 1)) = 2 (i.e. Petersen minus an edge)

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Construction of graphs with Φc ≥ 5

Theorem (Esperet,M.,Tarsi 2015) Let C be an odd cycle in a graph, along vertices of degree 3. The graph H obtained by a replacement of all edges of C with graphs

f capacity A, with Me(A) = 2, then φc(H) ≥ 5.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Construction of graphs with Φc ≥ 5

Theorem (Esperet,M.,Tarsi 2015) Let C be an odd cycle in a graph, along vertices of degree 3. The graph H obtained by a replacement of all edges of C with graphs

f capacity A, with Me(A) = 2, then φc(H) ≥ 5.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Construction of graphs with Φc ≥ 5

Theorem (Esperet,M.,Tarsi 2015) Let C be an odd cycle in a graph, along vertices of degree 3. The graph H obtained by a replacement of all edges of C with graphs

f capacity A, with Me(A) = 2, then φc(H) ≥ 5.

Me(CP (G ))=2

u,v

Gu,v

5

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Construction of graphs with Φc ≥ 5

Theorem (Esperet,M.,Tarsi 2015) Let C be an odd cycle in a graph, along vertices of degree 3. The graph H obtained by a replacement of all edges of C with graphs

f capacity A, with Me(A) = 2, then φc(H) ≥ 5.

Me(CP (G ))=2

u,v

Gu,v

5

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Some examples

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Some examples

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Some examples

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Some examples

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Snarks with φc ≥ 5

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A new construction (Mattiolo,M. 2017)

ODD CYCLE

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A new construction (Mattiolo,M. 2017)

= (1,4)−edge = (4,1)−edge

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A new construction (Mattiolo,M. 2017)

= (1,4)−edge = (4,1)−edge

CIRCULAR FLOW NUMBER (AT LEAST) 5

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

Some remarks: All known costructions produce a lot of redundancy; The main reason is that expansion operations and starting graph are (almost) completely arbitrary; Our main aim is a more compact and easy description in order to reduce redundancy.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

= (1,4)−edge

= (4,1)−edge Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

= (1,4)−edge

= (4,1)−edge Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

= (1,4)−edge = (4,1)−edge Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

= (1,4)−edge

= (4,1)−edge Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

Main problem: INPUT: Wn + even subgraph C of Wn + a set A ∈ SI5 of measure 2 . PROBLEM: Establish if the graph (Wn, CA) (obtained by replacing each edge of Wn in the even subgraph C with a generalized A-edge) has circular flow number (at least) 5.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

Main problem: INPUT: Wn + even subgraph C of Wn + a set A ∈ SI5 of measure 2 . PROBLEM: Establish if the graph (Wn, CA) (obtained by replacing each edge of Wn in the even subgraph C with a generalized A-edge) has circular flow number (at least) 5.

= (1,4)−edge = A−edge Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

Main problem: INPUT: Wn + even subgraph C of Wn + a set A ∈ SI5 of measure 2 . PROBLEM: Establish if the graph (Wn, CA) (obtained by replacing each edge of Wn in the even subgraph C with a generalized A-edge) has circular flow number (at least) 5.

= (1,4)−edge = A−edge

WARNING: THE ANSWER DEPENDS ON THE SET A

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

Definition A description of the even subgraph C in Wn (BLOCKS and CONNECTORS):

= (1,4)−edge

= A−edge

5−BLOCK 2−BLOCK

1−CONNECTOR 0−CONNECTOR

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

Only three elements of SI5 have measure 2:

(4, 1) (4, 0) ∪ (0, 1) (1, 2) ∪ (3, 4).

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

A unified description

Complete characterization: Theorem (Mattiolo,M. 2017) (Wn, CA) has circular flow number at least 5 IFF n ODD, and either A ∈ {(4, 1), (4, 0) ∪ (0, 1)} and C has not k-connectors with k ≥ 2

r

A = (1, 2) ∪ (3, 4) and C is the outer n-cycle of Wn.

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

Computational Results - joint work with J.Goedgebeur

SNARKS WITH CIRCULAR FLOW NUMBER (at least) 5 Order Number of snarks 10 1 12 . . . . . . 26 28 1 30 2 32 9 34 25 ALL OF THEM FALL INTO PREVIOUS DESCRIPTION (with suitable expansions of (W3, C(4,1)))

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

20th COMBINATORICS (Arco di Trento - Italy - June 3-9, 2018)

SEE YOU IN ITALY!

For information: www.combinatorics.it

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5

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Introduction Cubic graphs Open k-capacity Generation of graphs with circular flow number ≥ 5

THANKS FOR YOUR ATTENTION!

Giuseppe Mazzuoccolo Construction of snarks with circular flow number 5