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 � � f ( e ) = f ( e ) , e ∈ E + ( v ) e ∈ E − ( v ) 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 � � f ( e ) ≡ f ( e ) ( mod k ) , e ∈ E + ( v ) e ∈ E − ( v ) for each vertex v ∈ V ( G ). That is, ( D , f ) is a Z k -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 K 4 . 1 2 1 1 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 order 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 2 2 1 1 1 1 2 2 − 3 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 only 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 (2 k − 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 k 1 , k 2 be two integers. If G admits a k 1 -flow ( D , f 1 ) and a k 2 -flow ( D , f 2 ) such that supp ( f 1 ) ∪ supp ( f 2 ) = E ( G ) , then both ( D , k 2 f 1 + f 2 ) and ( D , f 1 + k 1 f 2 ) are nowhere-zero k 1 k 2 -flows of G. Example: 0 1 0 1 1 1 1 1 1 0 0 1 ( , ) ( , ) D f D f 1 2 2 1 1 3 1 2 ( ,2 ) D f f 1 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¨ otzch’s 3-coloring theorem Every bridgeless planar graph without 3-edge-cut is 3-face-colorable. As generalization of Gr¨ otzch’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 ⌈ log 2 n o ⌉ -edge-connected graph with at most n o vertices of odd-degree admits a nowhere-zero 3 -flow. Theorem (Alon, et.al. 1991) Every 2 ⌈ log 2 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 S d be the set of unit vectors on R d +1 . Let R k denote the set of k roots of unity. Definition ( S d -flow and R k -flow) An S d -flow is a flow whose flow values are vectors in S d . Similarly, an R k -flow is a flow whose flow values are vectors in R k . 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 R 3 -flow and every 14 -edge-connected graph has an R 5 -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
Recommend
More recommend
