Circular Flows of Graphs

Transcription

1 Circular Flows of Graphs Hong-Jian Lai and Cunquan Zhang West Virginia University Rui Xu University of West Georgia Circular Flows of Graphs p. 1/16

2 Notation: G: = a graph, with vertex set V = V (G) = {v 1, v 2,, v n }, and edge set E = E(G) = {e 1, e 2,, e m }. Circular Flows of Graphs p. 2/16

3 Notation: G: = a graph, with vertex set V = V (G) = {v 1, v 2,, v n }, and edge set E = E(G) = {e 1, e 2,, e m }. D(G): = an orientation of G. Circular Flows of Graphs p. 2/16

4 Notation: G: = a graph, with vertex set V = V (G) = {v 1, v 2,, v n }, and edge set E = E(G) = {e 1, e 2,, e m }. D(G): = an orientation of G. D = (d ij ) n m := vertex-edge incidence matrix, where d ij = 1 if e j is oriented away from v i 1 if e j is oriented into v i 0 otherwise Circular Flows of Graphs p. 2/16

5 Notation A function f : E A can be viewed as an m-dimensional vector f = (f(e 1 ), f(e 2 ),, f(e m )) T. Circular Flows of Graphs p. 3/16

6 Notation A function f : E A can be viewed as an m-dimensional vector f = (f(e 1 ), f(e 2 ),, f(e m )) T. A function b : V A can be viewed as an n-dimensional vector b = (b(v 1 ), b(v 2 ),, b(v n )) T. Circular Flows of Graphs p. 3/16

7 Notation A function f : E A can be viewed as an m-dimensional vector f = (f(e 1 ), f(e 2 ),, f(e m )) T. A function b : V A can be viewed as an n-dimensional vector b = (b(v 1 ), b(v 2 ),, b(v n )) T. A: = an abelian (additive) group with identity 0, and with A 3. Circular Flows of Graphs p. 3/16

8 Notation A function f : E A can be viewed as an m-dimensional vector f = (f(e 1 ), f(e 2 ),, f(e m )) T. A function b : V A can be viewed as an n-dimensional vector b = (b(v 1 ), b(v 2 ),, b(v n )) T. A: = an abelian (additive) group with identity 0, and with A 3. A = A {0}: = set of all non zero elements of A. Circular Flows of Graphs p. 3/16

9 Nowhere-zero A-flows (or A-NZFs) Assumption: For any graph G, we assume that a fixed orientation D(G) of G is given. Notation: a A, 1 a = a, ( 1) a = a (additive inverse of a in A), and 0 a = 0 (additive identity of A) Circular Flows of Graphs p. 4/16

10 Nowhere-zero A-flows (or A-NZFs) Assumption: For any graph G, we assume that a fixed orientation D(G) of G is given. Notation: a A, 1 a = a, ( 1) a = a (additive inverse of a in A), and 0 a = 0 (additive identity of A) For any f F (G, A), the boundary of f is f := Df. That is, v i V, f(v i ) = Df(v i ), which is the v i th component of the vector Df. Circular Flows of Graphs p. 4/16

11 Nowhere-zero A-flows (or A-NZFs) Assumption: For any graph G, we assume that a fixed orientation D(G) of G is given. Notation: a A, 1 a = a, ( 1) a = a (additive inverse of a in A), and 0 a = 0 (additive identity of A) For any f F (G, A), the boundary of f is f := Df. That is, v i V, f(v i ) = Df(v i ), which is the v i th component of the vector Df. A function f F (G, A) is a nowhere-zero A-flow (or just an A-NZF) if Df = 0 (the all zero vector). Circular Flows of Graphs p. 4/16

12 Circular Flows of Graphs p. 5/16 Integer Flows Z: = the abelian group of integers.

13 Circular Flows of Graphs p. 5/16 Integer Flows Z: = the abelian group of integers. Z k : = the abelian group of mod k integers.

14 Integer Flows Z: = the abelian group of integers. Z k : = the abelian group of mod k integers. A function f F (G, Z) is a nowhere-zero k-flow (or just a k-nzf) if Df = 0, and if e E(G), 0 < f(e) < k. Circular Flows of Graphs p. 5/16

15 Integer Flows Z: = the abelian group of integers. Z k : = the abelian group of mod k integers. A function f F (G, Z) is a nowhere-zero k-flow (or just a k-nzf) if Df = 0, and if e E(G), 0 < f(e) < k. Tutte: If G has a k-nzf, then G has a (k + 1)-NZF. Circular Flows of Graphs p. 5/16

16 Integer Flows Z: = the abelian group of integers. Z k : = the abelian group of mod k integers. A function f F (G, Z) is a nowhere-zero k-flow (or just a k-nzf) if Df = 0, and if e E(G), 0 < f(e) < k. Tutte: If G has a k-nzf, then G has a (k + 1)-NZF. Tutte: A graph G has an A-NZF if and only if G has an A -NZF. Circular Flows of Graphs p. 5/16

17 Some Properties If some orientation D(G) has an A-NZF or a k-nzf, then for any orientation of G also has the same property, and so having an A-MZF or a k-nzf is independent of the choice of the orientation. Circular Flows of Graphs p. 6/16

18 Some Properties If some orientation D(G) has an A-NZF or a k-nzf, then for any orientation of G also has the same property, and so having an A-MZF or a k-nzf is independent of the choice of the orientation. If for an abelian group A, a connected graph G has an A-NZF, then G must be 2-edge-connected. (That is, G does not have a cut edge). Circular Flows of Graphs p. 6/16

19 Some Properties If some orientation D(G) has an A-NZF or a k-nzf, then for any orientation of G also has the same property, and so having an A-MZF or a k-nzf is independent of the choice of the orientation. If for an abelian group A, a connected graph G has an A-NZF, then G must be 2-edge-connected. (That is, G does not have a cut edge). We shall only consider 2-edge-connected graphs G and define Λ(G) = min{k : G has a k NZF }. Circular Flows of Graphs p. 6/16

20 Circular Flows of Graphs p. 7/16 Tutte s Conjectures (5-flow) If κ (G) 2, then Λ(G) 5.

21 Tutte s Conjectures (5-flow) If κ (G) 2, then Λ(G) 5. (4-flow) If κ (G) 2, and if G does not have subgraph contractible to P 10, the Petersen graph, then Λ(G) 4. Circular Flows of Graphs p. 7/16

22 Tutte s Conjectures (5-flow) If κ (G) 2, then Λ(G) 5. (4-flow) If κ (G) 2, and if G does not have subgraph contractible to P 10, the Petersen graph, then Λ(G) 4. (3-flow) If κ (G) 4, then Λ(G) 3. Circular Flows of Graphs p. 7/16

23 Tutte s Conjectures (5-flow) If κ (G) 2, then Λ(G) 5. (4-flow) If κ (G) 2, and if G does not have subgraph contractible to P 10, the Petersen graph, then Λ(G) 4. (3-flow) If κ (G) 4, then Λ(G) 3. (Jaeger s weak 3-flow conjecture) There exists an integer k > 0 such that every k-edge-connected graph has a 3-NZF. Circular Flows of Graphs p. 7/16

24 Nowhere zero flows and colorings Tutte: For a plane graph G, G has a face k-coloring if and only if G has a k-nzf. Circular Flows of Graphs p. 8/16

25 Nowhere zero flows and colorings Tutte: For a plane graph G, G has a face k-coloring if and only if G has a k-nzf. These conjectures are theorems when restricted to planar graphs (need 4 Color Theorem for the 4-flow conjecture). Circular Flows of Graphs p. 8/16

26 What do we know? Jaeger (1979): Every 2-edge-connected graph has a 8-NZF. Circular Flows of Graphs p. 9/16

27 What do we know? Jaeger (1979): Every 2-edge-connected graph has a 8-NZF. Jaeger (1979): Every 4-edge-connected graph has a 4-NZF. Circular Flows of Graphs p. 9/16

28 What do we know? Jaeger (1979): Every 2-edge-connected graph has a 8-NZF. Jaeger (1979): Every 4-edge-connected graph has a 4-NZF. Seymour (1981): Every 2-edge-connected graph has a 6-NZF. Circular Flows of Graphs p. 9/16

29 What do we know? Jaeger (1979): Every 2-edge-connected graph has a 8-NZF. Jaeger (1979): Every 4-edge-connected graph has a 4-NZF. Seymour (1981): Every 2-edge-connected graph has a 6-NZF. The 5-flow conjecture and 3-flow conjecture have also been verified for projective planes and some other surfaces. Circular Flows of Graphs p. 9/16

30 What do we know? Robertson, Sanders, Seymour, Thomas (2000): Every 2-edge-connected cubic graph without a subgraph contractible to the Petersen graph has a 4-NZF. Circular Flows of Graphs p. 10/16

31 What do we know? Robertson, Sanders, Seymour, Thomas (2000): Every 2-edge-connected cubic graph without a subgraph contractible to the Petersen graph has a 4-NZF. HJL and C. Q. Zhang (1992): Every 4log 2 ( V (G) )-edge-connected graph has a 3-NZF. Circular Flows of Graphs p. 10/16

32 What do we know? Robertson, Sanders, Seymour, Thomas (2000): Every 2-edge-connected cubic graph without a subgraph contractible to the Petersen graph has a 4-NZF. HJL and C. Q. Zhang (1992): Every 4log 2 ( V (G) )-edge-connected graph has a 3-NZF. HJL, Y. Shao, H. Wu, J. Zhou (2007): Every 3log 2 ( V (G) )-edge-connected graph has a 3-NZF. Circular Flows of Graphs p. 10/16

33 What do we know? Robertson, Sanders, Seymour, Thomas (2000): Every 2-edge-connected cubic graph without a subgraph contractible to the Petersen graph has a 4-NZF. HJL and C. Q. Zhang (1992): Every 4log 2 ( V (G) )-edge-connected graph has a 3-NZF. HJL, Y. Shao, H. Wu, J. Zhou (2007): Every 3log 2 ( V (G) )-edge-connected graph has a 3-NZF. Z. H. Chen, H. Y. Lai and HJL (2002): Tutte s flow conjectures are valid if and only if they are valid within line graphs. Circular Flows of Graphs p. 10/16

34 Circular Flows of Graphs Let D = D(G) be an orientation of G. X V (G), δ + (X) = set of edges going out from X, and δ (X) = set of edges coming into X, and δ(x) = δ + (X) δ (X). Jaeger showed if φ c (G) = min D(G) max X V (G) δ(x) δ + (X), then Λ(G) = φ c (G). Therefore, φ c (G), known as the circular flow number of G, is a refinement of Λ(G), the flow number of G. Circular Flows of Graphs p. 11/16

35 Circular Flows of Graphs Question: If edge connectivity is high? will φ c (G) be less than 4? Circular Flows of Graphs p. 12/16

36 Circular Flows of Graphs Question: If edge connectivity is high? will φ c (G) be less than 4? Theorem: (A. Galluccio and L. A. Goddyn (2002), and R. Xu, C. Q. Zhang and HJL (2007)) If κ (G) 6, then φ c (G) < 4. Circular Flows of Graphs p. 12/16

37 Circular Flows of Graphs Question: If edge connectivity is high? will φ c (G) be less than 4? Theorem: (A. Galluccio and L. A. Goddyn (2002), and R. Xu, C. Q. Zhang and HJL (2007)) If κ (G) 6, then φ c (G) < 4. Theorem: Jaeger (1979): Every 4-edge-connected graph has a 4-NZF. Circular Flows of Graphs p. 12/16

38 Circular Flows of Graphs: Proof G has a 4-NZF g : E(G) {1, 2, 3} such that g(v) = 0, v V (G), under a given orientation D. Circular Flows of Graphs p. 13/16

39 Circular Flows of Graphs: Proof G has a 4-NZF g : E(G) {1, 2, 3} such that g(v) = 0, v V (G), under a given orientation D. Suppose that φ c (G) = 4. Then for some X V (G), δ + (X) + δ (X) = δ(x) < 4 δ + (X). Circular Flows of Graphs p. 13/16

40 Circular Flows of Graphs: Proof G has a 4-NZF g : E(G) {1, 2, 3} such that g(v) = 0, v V (G), under a given orientation D. Suppose that φ c (G) = 4. Then for some X V (G), δ + (X) + δ (X) = δ(x) < 4 δ + (X). This implies that g 1 (2) δ(x) =. Circular Flows of Graphs p. 13/16

41 Circular Flows of Graphs: Proof G has a 4-NZF g : E(G) {1, 2, 3} such that g(v) = 0, v V (G), under a given orientation D. Suppose that φ c (G) = 4. Then for some X V (G), δ + (X) + δ (X) = δ(x) < 4 δ + (X). This implies that g 1 (2) δ(x) =. Idea: If we can find a positive 4-NZF f, such that f 1 (2) δ(x), then we must have Λ(G) < 4. Circular Flows of Graphs p. 13/16

42 Circular Flows of Graphs: Proof By Nash-Williams and Tutte, κ (G) 6 implies G has three edge-disjoint spanning trees T 1, T 2 and T 3. Circular Flows of Graphs p. 14/16

43 Circular Flows of Graphs: Proof By Nash-Williams and Tutte, κ (G) 6 implies G has three edge-disjoint spanning trees T 1, T 2 and T 3. Find an O(G)-joint P i in T i. (An O(G)-joint is a subgraph whose set of odd degree vertices is O(G), the set of all odd degree vertices of G). Circular Flows of Graphs p. 14/16

44 Circular Flows of Graphs: Proof By Nash-Williams and Tutte, κ (G) 6 implies G has three edge-disjoint spanning trees T 1, T 2 and T 3. Find an O(G)-joint P i in T i. (An O(G)-joint is a subgraph whose set of odd degree vertices is O(G), the set of all odd degree vertices of G). Let C 1 = P 1 P 2, and C 2 = G E(P 1 ). Then both are even subgraphs (graphs with components being eulerian). Let f 1 : C 1 {1} and f 2 : C 2 {2}, and f = f 1 + f 2 : E(G) {1, 2, 3} such that T 3 f 1 (2). Circular Flows of Graphs p. 14/16

45 New Conjectures What we have proved: If E(G) can be partitioned into J 1 J 2 L 1 such that each J i and L j is an O(G)-joint, and such that each L j is spanning and connected, then Λ(G) < 4. Circular Flows of Graphs p. 15/16

46 New Conjectures What we have proved: If E(G) can be partitioned into J 1 J 2 L 1 such that each J i and L j is an O(G)-joint, and such that each L j is spanning and connected, then Λ(G) < 4. New Conjecture: If E(G) can be partitioned into J 1 J 2... L t L 1 L 2... L k such that each J i and L j is an O(G)-joint, and such that each L j is spanning and connected, and if t 2 and k 1, then Λ(G) 2 + k+1 k. Circular Flows of Graphs p. 15/16

47 Thank You! Circular Flows of Graphs p. 16/16