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Modular orientations

Louis Esperet

CNRS, Laboratoire G-SCOP, Grenoble, France

Second GATO meeting, GrenobleJanuary 16, 2018

Orientations in graphs

Given a function α : V → N, an α-orientation of a graph G = (V ,E ) is anorientation of G in which each vertex v ∈ G has out-degree α(v).

Necessary conditions for the existence of an α-orientation:

(i)∑

v∈V α(v) = m (m is the number of edges of G )

(ii) ∀A ⊆ V , |E (A)| ≤∑

v∈A α(v) ≤ |E (A)|+ |∂A| (E (A) is the set of edgeswith both endpoints in A, and ∂A is the set of edges with exactly oneendpoint in A).

These conditions are also sufficient (by reduction to a flow problem).

What about the existence of modular α-orientations? Can we avoid one of thetwo inequalities?

(i)∑

(ii) ∀A ⊆ V , |E (A)| ≤∑

(i)∑

(ii) ∀A ⊆ V , |E (A)| ≤∑

(i)∑

(ii) ∀A ⊆ V , |E (A)| ≤∑

(i)∑

(ii) ∀A ⊆ V , |E (A)| ≤∑

(i)∑

(ii) ∀A ⊆ V , |E (A)| ≤∑

Flows in graphs

A k-flow in an oriented graph G is an assignment of integers of ±{0, 1, . . . , k − 1}to the arcs of G , such that for each vertex v , the sum of the values of the arcsentering v is equal to the sum of the values of the arcs leaving v .

We say that the flow is nowhere-zero if every arc has a non-zero value.

Orientations are irrelevant in this problem.

For plane graphs, flows are in duality with proper vertex colorings.

Any 2-edge-connected graph has a nowhere-zero 5-flow.

Conjecture (Tutte 1954)

Flows in graphs

Group-valued flows

Instead of integers in ±{1, . . . , k − 1}, we can use the non-zero elements of anyabelian group (A,+). In this case we talk about nowhere-zero A-flow.

For any integer k and any abelian group A of cardinality k, any graph has anowhere-zero k-flow if and only if it has a nowhere-zero A-flow.

Theorem (Tutte 1954)

In particular:

G has a nowhere-zero k-flow ⇔ G has a nowhere-zero Zk -flow

G has a nowhere-zero 2k -flow ⇔ G has a nowhere-zero Zk2-flow

⇔ the edge-set of G can be covered by k Eulerian subgraphs

Group-valued flows

In particular:

Group-valued flows

In particular:

Group-valued flows

In particular:

Group-valued flows

In particular:

Group-valued flows

In particular:

4-edge-connected graphs

Every 4-edge-connected graph has a nowhere-zero Z3-flow.

Theorem (Jaeger 1979)

Let G = (V ,E ) be a graph. For any spanning tree T , the graph G has anEulerian subgraph containing all the edges of E \ T .

Every 2k-edge-connected graph contains k edge-disjoint spanning trees.

Theorem (Nash-Williams 1961)

Highly edge-connected graphs

Every 4k-edge-connected graph has an orientation together with a Z2k+1-flowin which each arc is assigned 1 (mod 2k + 1).

Conjecture (Jaeger 1979)

Every 6k-edge-connected graph has an orientation together with a Z2k+1-flow inwhich each arc is assigned 1 (mod 2k+1). In particular every 6-edge-connectedgraph has a nowhere-zero 3-flow.

Theorem (Lovasz, Thomassen, Wu, Zhang 2013)

For the sake of induction, they prove a stronger result, where each vertex v startswith some value β(v) ∈ Z2k+1, and they want that for every vertex v , theout-degree of v minus the in-degree of v is equal to β(v) (mod 2k + 1).

Additive basis of vector spaces

Additive basis over a vector space F : a multiset B of elements of F such that anyelement of F can be expressed as a sum of subset of elements of B.

If p is a prime and A,B are two non-empty subsets of Zp, then |A + B| ≥min(p, |A|+ |B| − 1).

Theorem (Cauchy-Davenport 1935)

For any prime p, any multiset of p−1 non-zero elements of Zp forms an additivebasis of Zp.

Corollary

Generalization to arbitrary vector spaces over prime fields?

Corollary

Additive basis over vector spaces

For any prime p there is a constant c(p) such that for any integer n, the union(with repetition) of any c(p) linear bases of Zn

p forms an additive basis of Znp.

Conjecture (Jaeger, Linial, Payan, Tarsi 1992)

For any prime p and any integer n, the union (with repetition) of any pdlog nelinear bases of Zn

p contains an additive basis of Znp.

Theorem (Alon, Linial, Meshulam 1991)

The support of a vector x = (x1, . . . , xn) ∈ Znp is the set of indices i such that

xi 6= 0.

For any prime p and any integer n, the union (with repetition) of any 12p3

linear bases of Znp such that the support of each vector has size at most 2,

forms an additive basis of Znp.

Theorem (E., de Joannis de Verclos, Le, Thomasse 2017)

xi 6= 0.

Connection with additive bases

Consider a highly edge-connected graph G = (V ,E ) on n vertices, with anarbitrary orientation. We construct a matrix M(G ) over Z2k+1 as follows:

the rows of M(G ) are indexed by the vertices of G , and

for each arc uv , we add a column in which the element indexed by u is 1, theelement indexed by v is −1, and all the other elements are 0.

G is highly edge-connected ⇔ G has many disjoint spanning trees⇒ M(G ) contains many disjoint linear bases of Zn

⇒ M(G ) contains an additive basis of Zn2k+1

⇒ For any β : V → Z2k+1, G has a subgraph H such that for any vertex v , theout-degree of v in H minus the in-degree of v in H is equal to β(v) (mod 2k + 1)

⇔ For any β : V → Z2k+1, G has an orientation such that for any vertex v , theout-degree of v minus the in-degree of v is equal to β(v) (mod 2k + 1)

G is highly edge-connected

⇔ G has many disjoint spanning trees⇒ M(G ) contains many disjoint linear bases of Zn

G is highly edge-connected ⇔ G has many disjoint spanning trees

⇒ M(G ) contains many disjoint linear bases of Zn2k+1

Back to flows

Let p be a prime number, and let G be a 6p2-edge-connected graph with a fixedorientation. Assume that for each arc, we have a list of two possible values inZp. Then G has a Zp-flow in which each arc is assigned a value from its ownlist.

Back to flows

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