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Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Vector Flows and integer flows

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang

• Dec. 20, 2016, Shanghai

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

• 1. Introduction and basic properties

Definition Let G be a graph, D be an orientation of G, Γ be an abelian group, and f : E(G) → Γ be a mapping. Then the ordered pair (D, f ) is called a flow (or group Γ-flow) of G if

• e∈E +(v)

f (e) =

• e∈E −(v)

f (e), for each vertex v ∈ V (G). (D, f ) is called an integer flow if Γ = Z; An integer flow (D, f ) is an integer k-flow if |f (e)| < k for each edge e ∈ E(G).

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

(D, f ) is called a modulo-k-flow if f : E(G) → Z, such that

• e∈E +(v)

f (e) ≡

• e∈E −(v)

f (e) (mod k), for each vertex v ∈ V (G). That is, (D, f ) is a Zk-flow. The Support of a Γ-flow (D, f ) is the set of all edges of G with f (e) = 0, is denoted by supp(f ). A flow (D, f ) is nowhere-zero if supp(f ) = E(G). For example, a nowhere-zero integer 4-flow in the K4. 1 1 1 2 2 3

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Two important properties

Tutte, 1949 Let G be a graph, k be a positive integer, and Γ be an abelian group of

• rder k. Then the following statements are equivalent:

(a) G admits a nowhere-zero integer k-flow; (b) G admits a nowhere-zero modulo k-flow; (c) G admits a nowhere-zero group Γ-flow. Existence of flow is independent on the orientation D of G. 1 1 1 2 2 3 1 1 1 2 2 3 −

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Motivation—–Flow-coloring duality

The following theorem of Tutte indicate the important relation between map coloring and integer flows, motivated and promotes the study of the theory of integer flow. Theorem (Tutte, 1954) Let G be a planar bridgeless graph. Then G is k-face-colorable if and

• nly if G admits a nowhere-zero k-flow.

Four color Theorem Every bridgeless planar graph admits a nowhere-zero 4-flow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

“only if ” part of Tutte’s Theorem holds not only for planar graphs, but also for all graphs embeddable on some orientable surface.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Three Famous Conjectures

5-flow Conjecture Every bridgeless graph admits a nowhere-zero 5-flow. 4-flow Conjecture Every bridgeless graph containing no subdivision of the Petersen graph admits a nowhere-zero 4-flow. 3-flow Conjecture Every 4-edge connected graph admits a nowhere-zero 3-flow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Tool 1: Modulo k-orientation

Definition Let k be an odd integer. An orientation D of a graph G is called a modulo k-orientation if d+

D (v) ≡ d− D (v) (mod k),

for every v ∈ V (G). Theorem (Jaeger, 1984) G has a modulo 3-orientation if and only if G admits a nowhere-zero 3-flow. Modulo k-orientation Conjecture (Jeager, 1984) If G is (2k − 2)-edge-connected, then G has a modulo k-orientation.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Tool 2: Product of flows

Lemma Let G be a graph and k1, k2 be two integers. If G admits a k1-flow (D, f1) and a k2-flow (D, f2) such that supp(f1) ∪ supp(f2) = E(G), then both (D, k2f1 + f2) and (D, f1 + k1f2) are nowhere-zero k1k2-flows of G. Example:

1 1 1 1 1 1  1 1

1

( , ) D f

2

( , ) D f 3 1 1 1 2

1 2

( ,2 ) D f f  2

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

5-flow Conjecture and related problems

Three famous Theorems 5-color Theorem(Heawood, 1890) Every bridgeless planar graph admits a nowhere-zero 5-flow. 8-flow Theorem(Jaeger, 1979, JCTB) Every bridgeless graph admits a nowhere-zero 8-flow. 6-flow Theorem(Seymour, 1981, JCTB) Every bridgeless graph admits a nowhere-zero 6-flow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Two newest results

Theorem (Kochol, 2010, JCTB) A smallest counter-example G to the 5-flow conjecture has the following properties. (a) G has girth at least 11; (b) G has cyclic edge-connectivity at least 6. Conjecture (Jaeger, 1988, JCTB) Every 9-edge-connected graph has a modulo 5-orientation. Jaeger’s Conjecture implies the 5-flow Conjecture. Theorem (Thomassen, 2012, JCTB) Every 55-edge-connected graph has a modulo 5-orientation. Theorem (Lov´ asz, Thomassen, Wu, C.Q. Zhang, 2012, JCTB) Every 12-edge-connected graph has a modulo 5-orientation.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

3-flow Conjecture and related Problems

Gr¨

• tzch’s 3-coloring theorem

Every bridgeless planar graph without 3-edge-cut is 3-face-colorable. As generalization of Gr¨

• tzch’s 3-coloring theorem, Tutte proposed 3-flow

Conjecture, a major open problem in integer flow theory. 3-flow Conjecture Every bridgeless graph without 3-edge-cut admits a nowhere-zero 3-flow. Other equivalent version. Three 3-Cuts Conjecture(Kochol, 2002) Every bridgeless graph having at most three 3-cuts, each of which trivial, admits a nowhere-zero 3-flow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Weak 3-flow Conjecture

Weak 3-flow Conjecture(Jaeger, 1979, JCTB) There is an integer h for which every h-edge-connected graph admits a nowhere-zero 3-flow. Theorem (H.-J. Lai, C.-Q. Zhang, 1992) Every 4⌈log2 no⌉-edge-connected graph with at most no vertices of

• dd-degree admits a nowhere-zero 3-flow.

Theorem (Alon, et.al. 1991) Every 2⌈log2 n⌉-edge-connected graph admits a nowhere-zero 3-flow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Weak 3-flow conjecture was proved by C. Thomassen. Theorem (Thomassen, 2012, JCTB) Every 8-edge-connected graph admits a nowhere-zero 3-flow. Theorem (Lov´ asz, Thomassen, Wu, C.-Q. Zhang, 2013, JCTB) Every 6-edge-connected graph admits a nowhere-zero 3-flow. Note that Kochol (2001) proved the minimum counter-example is 5-edge-connected; C.Q. Zhang(2002) proved the minimum counter-example is 5-regular and 5-odd-edge-connected. So, above Theorem is just one step away from resolution.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Vector flow

Let Sd be the set of unit vectors on Rd+1. Let Rk denote the set of k roots of unity. Definition (Sd-flow and Rk-flow) An Sd-flow is a flow whose flow values are vectors in Sd. Similarly, an Rk-flow is a flow whose flow values are vectors in Rk. Recently, C. Thomassen discussed 3-flow Conjecture and 5-flow Conjecture in view of vector flow and proved the following Theorem. Theorem (Thomassen, 2014, JCTB) Every 6-edge-connected graph admits an R3-flow and every 14-edge-connected graph has an R5-flow. If the edge connectivity 6 and 14 can be reduced to 4 and 9, respectively, then 3-flow Conjecture and 5-flow Conjecture follow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Conjecture (Jain, 2007) Every 4-edge-connected graph has an S1-flow and every 2-edge-connected graph has an S2-flow. The first part of the Conjecture indicates that there is a very close relationship between nowhere-zero 3-flow and S1-flow. DeVos(2007) also pointed out that “it seems likely that a graph has an S1-flow if and only if it has a nowhere-zero 3-flow”. Thomassen constructed a class of counterexamples, and show suggested equivalence holds when S1 is replaced by R3.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Theorem (Thomassen, 2014, JCTB) Let G be a planar graph. Then G admits a vector S1-flow if and only if the dual graph G ∗ is homomorphic to a subgraph of unit distance graph, where unit distance graph is the graph whose vertices are points in Euclidian plane R2 such that two vertices are adjacent if and only if the have distance 1.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Statement A. G admits a nowhere zero 3-flow. Statement B. G admits a S1-flow. Statement C. G admits a R3-flow. Theorem (C.Thomassen, 2014, JCTB) (1) Statements A and C are equivalent. (2) Statements A implies statement B, but not visa versa. (3) Statements A and B are equivalent for cubic graphs.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Due to the existence of such counterexample, it is natural to ask the following two problems. Problems 1 Characterize graphs that Statement B implies Statement A. Problems 2 Characterize vector S1-flow that Statement B implies Statement A.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Our results

Theorem (Wang, Chen, Luo, Zhang, 2015, SIAM DM) Let G be a graph such that (1) V3 = ∅ and G[V3] is connected; (2) G − V3 is acyclic. Then G has a nowhere zero 3-flow if and only if it has an S1-flow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

❤ ❤ ❤ ❤ ❤ ❤ ❍❍❍ ❍ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ✟ ✟ ✟ ✟ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✟✟✟ ✟ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ❍ ❍ ❍ ❍ ❏ ❏ ❏ ❏ ❏ ❏ ❏

Fig.1 Thomassen’s counterexample G

Note that G[V3] has two components. So, the condition (1) in above Theorem might be not dropped !

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Theorem (Wang, Chen, Luo, Zhang, 2015, SIAM DM) G has a nowhere-zero 3-flow if and only if G has an S1

3-flow, where

S1

3-flow is the same as a flow whose flow values are 3 vectors of S1.

Let (D, f ) be a vector S1-flow with flow values {α1, α2, . . . , αb}, where {α1, α2, . . . , αb} consists of b pairwise non-collinear vector of S1. For each vertex v,

• ǫ(v) = (ǫ1(v), ǫ2(v), . . . , ǫb(v)) is called a balanced

vector of (D, f ), where ǫi(v) = ǫ+

i (v) − ǫ− i (v),

ǫ+

i (v) = ♯{e ∈ E +(v) :

f (e) = αi}, ǫ−

i (v) = ♯{e ∈ E −(v) :

f (e) = αi}. The rank of (D, f ) is defined as the rank of linear space generated by all balanced vectors.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Note that there exists at least one counterexample given by Thomassen with rank k for all integer k ≥ 3. Theorem (Wang, Chen, Luo, Zhang, 2015, SIAM DM) If G admits an S1-flow with rank at most 2, then G has a nowhere-zero 3-flow.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

Further work

If G admits an S1-flow with rank 3, then G admits a nowhere-zero 4-flow? Explore relationships between admitting an S2-flow and admitting a nowhere-zero 5-flow. Understand the flow from linear algebra viewpoint and characterize the flow by linear algebra tools.

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows

Introduction to integer flow 5-flow Conjecture and 3-flow Conjecture Vector flow

The End

Thank you!

Yi Wang School of mathematical sciences, Anhui University Joint with Jian Cheng, Rong Luo and Cun-Quan Zhang Vector Flows and integer flows