nowhere zero 3 flows in cayley graphs of nilpotent groups
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Nowhere-zero 3-flows in Cayley graphs of nilpotent groups Martin Skoviera Comenius University, Bratislava Maps and Riemann Surfaces Institute of Mathematics, RAS, Novosibirsk 3rd November, 2014 Martin Skoviera (Bratislava) Flows in

  1. Nowhere-zero 3-flows in Cayley graphs of nilpotent groups Martin ˇ Skoviera Comenius University, Bratislava Maps and Riemann Surfaces Institute of Mathematics, RAS, Novosibirsk 3rd November, 2014 Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 1 / 23

  2. Nowhere-zero flows A nowhere-zero k-flow on a graph G is an assignment of a direction and a value from {± 1 , ± 2 , . . . , ± ( k − 1) } to each edge of G so that at each vertex flow in = flow out. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 2 / 23

  3. Nowhere-zero flows A nowhere-zero k-flow on a graph G is an assignment of a direction and a value from {± 1 , ± 2 , . . . , ± ( k − 1) } to each edge of G so that at each vertex flow in = flow out. G must be bridgeless Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 2 / 23

  4. Nowhere-zero flows A nowhere-zero k-flow on a graph G is an assignment of a direction and a value from {± 1 , ± 2 , . . . , ± ( k − 1) } to each edge of G so that at each vertex flow in = flow out. G must be bridgeless reverse edge-direction + opposite flow-value = same flow Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 2 / 23

  5. Nowhere-zero flows A nowhere-zero k-flow on a graph G is an assignment of a direction and a value from {± 1 , ± 2 , . . . , ± ( k − 1) } to each edge of G so that at each vertex flow in = flow out. G must be bridgeless reverse edge-direction + opposite flow-value = same flow k -flow ⇒ ( k + 1)-flow Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 2 / 23

  6. Nowhere-zero flows A nowhere-zero k-flow on a graph G is an assignment of a direction and a value from {± 1 , ± 2 , . . . , ± ( k − 1) } to each edge of G so that at each vertex flow in = flow out. G must be bridgeless reverse edge-direction + opposite flow-value = same flow k -flow ⇒ ( k + 1)-flow Question What is the smallest value of m for which G has a nowhere-zero m -flow? Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 2 / 23

  7. Group-valued flows Let A be an abelian group. A nowhere-zero A-flow on a graph G is an assignment of a direction and a value from A − 0 to each edge of G so that at each vertex flow in = flow out. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 3 / 23

  8. Group-valued flows Let A be an abelian group. A nowhere-zero A-flow on a graph G is an assignment of a direction and a value from A − 0 to each edge of G so that at each vertex flow in = flow out. Theorem (Tutte, 1950) For every graph G the following statements are equivalent. G has a nowhere-zero k-flow. G has a nowhere-zero Z k -flow. G has a nowhere-zero A-flow, where | A | = k. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 3 / 23

  9. � � ��� ��� Flows and face-colourings Theorem (Tutte, 1949) Let K be a graph 2 -cell embedded in an orientable surface S. If the embedding is m-face-colourable, then K admits a nowhere-zero m-flow. If S is the 2 -sphere, the converse holds as well. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 4 / 23

  10. Tutte’s flow conjectures 5-Flow Conjecture (1954): Every bridgeless graph admits a nowhere-zero 5-flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 5 / 23

  11. Tutte’s flow conjectures 5-Flow Conjecture (1954): Every bridgeless graph admits a nowhere-zero 5-flow. 4-Flow Conjecture (1966): Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 5 / 23

  12. Tutte’s flow conjectures 5-Flow Conjecture (1954): Every bridgeless graph admits a nowhere-zero 5-flow. 4-Flow Conjecture (1966): Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow. 3-Flow Conjecture (1972): Every bridgeless graph with no 3-edge-cut has a nowhere-zero 3-flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 5 / 23

  13. Known results – 5FC Theorem (Jaeger, 1976) Every bridgeless graph has a nowhere-zero 8 -flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 6 / 23

  14. Known results – 5FC Theorem (Jaeger, 1976) Every bridgeless graph has a nowhere-zero 8 -flow. Theorem (Seymour, 1981) Every bridgeless graph has a nowhere-zero 6 -flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 6 / 23

  15. Known results – 5FC Theorem (Jaeger, 1976) Every bridgeless graph has a nowhere-zero 8 -flow. Theorem (Seymour, 1981) Every bridgeless graph has a nowhere-zero 6 -flow. 5-FC has been verified for various classes of graphs (but remains widely open) the conjecture reduces to verification on snarks (‘non-trivial’ cubic graphs that fail to have a 3-edge-colouring; equivalently, nowhere-zero 4-flow ) the smallest counterexample must be a cyclically 6-connected snark of girth ≥ 9 (Kochol, 2006) Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 6 / 23

  16. Known results – 4FC Petersen Minor Conjecture : Every bridgeless graph cubic graph with no Petersen minor is 3-edge-colourable. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 7 / 23

  17. Known results – 4FC Cubic 4-Flow Conjecture : Every bridgeless graph cubic graph with no Petersen minor has a nowhere-zero 4-flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 7 / 23

  18. Known results – 4FC Cubic 4-Flow Conjecture : Every bridgeless graph cubic graph with no Petersen minor has a nowhere-zero 4-flow. C4FC is equivalent to its restriction to a class of almost planar graphs consisting of 2-connected apex graphs and double-cross graphs [Robertson, Seymour, Thomas, 1997] Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 7 / 23

  19. Known results – 4FC Cubic 4-Flow Conjecture : Every bridgeless graph cubic graph with no Petersen minor has a nowhere-zero 4-flow. C4FC is equivalent to its restriction to a class of almost planar graphs consisting of 2-connected apex graphs and double-cross graphs [Robertson, Seymour, Thomas, 1997] The authors announced that they had proved the restricted conjecture, thereby establishing the C4FC. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 7 / 23

  20. Known results – 3FC Theorem (Jaeger, 1976) Every bridgeless graph with no 3 -edge-cut has a nowhere-zero 4 -flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 8 / 23

  21. Known results – 3FC Theorem (Jaeger, 1976) Every bridgeless graph with no 3 -edge-cut has a nowhere-zero 4 -flow. 3FC has been verified e.g. for projective planar graphs (Steinberg & Younger, 1989) Cartesian products (Imrich & ˇ Skrekovski, 2003; Shu & Zhang, 2005) random graphs (Sudakov, 2001) and reduced to 5-edge-connected 5-regular graphs (Zhang, Kochol, 2002). Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 8 / 23

  22. Known results – 3FC Theorem (Jaeger, 1976) Every bridgeless graph with no 3 -edge-cut has a nowhere-zero 4 -flow. 3FC has been verified e.g. for projective planar graphs (Steinberg & Younger, 1989) Cartesian products (Imrich & ˇ Skrekovski, 2003; Shu & Zhang, 2005) random graphs (Sudakov, 2001) and reduced to 5-edge-connected 5-regular graphs (Zhang, Kochol, 2002). Theorem (Thomassen, 2012) Every 8 -edge-connected graph admits a nowhere-zero 3 -flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 8 / 23

  23. Flows and symmetry in graphs Only two vertex-transitive graphs with no nowhere-zero 4-flow are known: Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 9 / 23

  24. Flows and symmetry in graphs Only two vertex-transitive graphs with no nowhere-zero 4-flow are known: Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 9 / 23

  25. Flows and symmetry in graphs Only two vertex-transitive graphs with no nowhere-zero 4-flow are known: Natural question: What is the effect of graph symmetry on the existence of nowhere-zero flows on graphs? Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 9 / 23

  26. Flows and symmetry in graphs Conjecture 1 (Lov´ asz, 1969) Every connected vertex-transitive graph has a hamilton path. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 10 / 23

  27. Flows and symmetry in graphs Conjecture 1 (Lov´ asz, 1969) Every connected vertex-transitive graph has a hamilton path. Conjecture 2 (folklore) Every Cayley graph (of valency ≥ 2) has a hamilton cycle. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 10 / 23

  28. Flows and symmetry in graphs Conjecture 1 (Lov´ asz, 1969) Every connected vertex-transitive graph has a hamilton path. Conjecture 2 (folklore) Every Cayley graph (of valency ≥ 2) has a hamilton cycle. Conjecture 3 (Alspach et al., 1996) Every Cayley graph (of valency ≥ 2) admits a nowhere-zero 4-flow. Martin ˇ Skoviera (Bratislava) Flows in Cayley graphs IM RAN, 03/11/2014 10 / 23

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