Nowhere-zero 3-flows in arc-transitive graphs on nilpotent groups Sanming Zhou Department of Mathematics and Statistics The University of Melbourne, Australia Monash University, October 28, 2013 Joint work with Xiangwen Li, Central China Normal University (Wuhan)
3-flows in arc-transitive circulations graphs Circulations Definition and integer flows Let D = ( V ( D ) , A ( D )) be a digraph and A an abelian group. Tutte’s flow A circulation in D over A is a function conjectures 3-flows in vertex- f : A ( D ) → A transitive graphs such that Proof � � f ( a ) = f ( a ) , for all v ∈ V ( D ) , a ∈ A + ( v ) a ∈ A − ( v ) where A + ( v ) ( A − ( v ), respectively) is the set of arcs of D leaving from v (entering into v , respectively).
3-flows in arc-transitive circulations graphs Circulations Definition and integer flows Let D = ( V ( D ) , A ( D )) be a digraph and A an abelian group. Tutte’s flow A circulation in D over A is a function conjectures 3-flows in vertex- f : A ( D ) → A transitive graphs such that Proof � � f ( a ) = f ( a ) , for all v ∈ V ( D ) , a ∈ A + ( v ) a ∈ A − ( v ) where A + ( v ) ( A − ( v ), respectively) is the set of arcs of D leaving from v (entering into v , respectively). We say that f is nowhere-zero if f ( a ) � = 0 for every a ∈ A ( D ), where 0 is the identity element of A .
3-flows in arc-transitive graphs Theorem Circulations (W. Tutte 1954) and integer flows A plane digraph is k-face-colorable if and only if it admits a Tutte’s flow nowhere-zero circulation over Z k . conjectures 3-flows in vertex- transitive graphs Proof
3-flows in arc-transitive graphs Theorem Circulations (W. Tutte 1954) and integer flows A plane digraph is k-face-colorable if and only if it admits a Tutte’s flow nowhere-zero circulation over Z k . conjectures 3-flows in vertex- transitive Whether a digraph admits a nowhere-zero circulation over a graphs given abelian group depends only on its underlying undirected Proof graph. So we can speak of nowhere-zero circulations in undirected graphs.
3-flows in arc-transitive graphs Theorem Circulations (W. Tutte 1954) and integer flows A plane digraph is k-face-colorable if and only if it admits a Tutte’s flow nowhere-zero circulation over Z k . conjectures 3-flows in vertex- transitive Whether a digraph admits a nowhere-zero circulation over a graphs given abelian group depends only on its underlying undirected Proof graph. So we can speak of nowhere-zero circulations in undirected graphs. Four-Color-Theorem Restated: Every planar graph admits a nowhere-zero circulation over Z 4 .
3-flows in arc-transitive integer flows graphs Circulations and integer Definition flows A nowhere-zero circulation f over Z in a digraph D is called a Tutte’s flow conjectures (nowhere-zero) k -flow if 3-flows in vertex- transitive − ( k − 1) ≤ f ( a ) ≤ k − 1 , for all a ∈ A ( D ) graphs Proof
3-flows in arc-transitive integer flows graphs Circulations and integer Definition flows A nowhere-zero circulation f over Z in a digraph D is called a Tutte’s flow conjectures (nowhere-zero) k -flow if 3-flows in vertex- transitive − ( k − 1) ≤ f ( a ) ≤ k − 1 , for all a ∈ A ( D ) graphs Proof Theorem (W. Tutte 1954) A graph admits a k-flow if and only if it admits a nowhere-zero circulation over Z k .
3-flows in arc-transitive integer flows graphs Circulations and integer Definition flows A nowhere-zero circulation f over Z in a digraph D is called a Tutte’s flow conjectures (nowhere-zero) k -flow if 3-flows in vertex- transitive − ( k − 1) ≤ f ( a ) ≤ k − 1 , for all a ∈ A ( D ) graphs Proof Theorem (W. Tutte 1954) A graph admits a k-flow if and only if it admits a nowhere-zero circulation over Z k . Four-Color-Theorem Again: Every planar graph admits a 4-flow.
3-flows in arc-transitive graphs Circulations and integer flows Tutte’s flow Theorem conjectures A graph admits a 2 -flow if and only if its vertices all have even 3-flows in vertex- degrees. transitive graphs Theorem Proof A 2 -edge-connected cubic graph admits a 3 -flow if and only if it is bipartite.
3-flows in arc-transitive Tutte’s 5-flow conjecture graphs Circulations and integer Tutte proposed three conjectures on integer flows (1954, 1968, flows 1972). Tutte’s flow conjectures 3-flows in vertex- transitive graphs Proof
3-flows in arc-transitive Tutte’s 5-flow conjecture graphs Circulations and integer Tutte proposed three conjectures on integer flows (1954, 1968, flows 1972). Tutte’s flow conjectures Conjecture 3-flows in vertex- transitive (The 5 -flow conjecture) graphs Every 2 -edge-connected graph admits a 5 -flow. Proof
3-flows in arc-transitive Tutte’s 5-flow conjecture graphs Circulations and integer Tutte proposed three conjectures on integer flows (1954, 1968, flows 1972). Tutte’s flow conjectures Conjecture 3-flows in vertex- transitive (The 5 -flow conjecture) graphs Every 2 -edge-connected graph admits a 5 -flow. Proof Theorem (The 8 -flow theorem, F. Jaeger 1976) Every 2 -edge-connected graph admits a 8 -flow.
3-flows in arc-transitive Tutte’s 5-flow conjecture graphs Circulations and integer Tutte proposed three conjectures on integer flows (1954, 1968, flows 1972). Tutte’s flow conjectures Conjecture 3-flows in vertex- transitive (The 5 -flow conjecture) graphs Every 2 -edge-connected graph admits a 5 -flow. Proof Theorem (The 8 -flow theorem, F. Jaeger 1976) Every 2 -edge-connected graph admits a 8 -flow. Theorem (The 6 -flow theorem, P. Seymour 1981) Every 2 -edge-connected graph admits a 6 -flow.
3-flows in arc-transitive Tutte’s 4-flow conjecture graphs Circulations and integer flows Conjecture Tutte’s flow conjectures (The 4 -flow conjecture) 3-flows in Every 2 -edge-connected graph with no Petersen graph minor vertex- transitive admits a 4 -flow. graphs Proof
3-flows in arc-transitive Tutte’s 4-flow conjecture graphs Circulations and integer flows Conjecture Tutte’s flow conjectures (The 4 -flow conjecture) 3-flows in Every 2 -edge-connected graph with no Petersen graph minor vertex- transitive admits a 4 -flow. graphs Proof Confirmed for cubic graphs by Robertson, Sanders, Seymour and Thomas.
3-flows in arc-transitive Tutte’s 4-flow conjecture graphs Circulations and integer flows Conjecture Tutte’s flow conjectures (The 4 -flow conjecture) 3-flows in Every 2 -edge-connected graph with no Petersen graph minor vertex- transitive admits a 4 -flow. graphs Proof Confirmed for cubic graphs by Robertson, Sanders, Seymour and Thomas. Theorem (F. Jaeger 1979) Every 4 -edge-connected graph admits a 4 -flow.
3-flows in arc-transitive Tutte’s 3-flow conjecture graphs Circulations and integer flows Tutte’s flow conjectures Conjecture 3-flows in vertex- (The 3 -flow conjecture) transitive graphs Every 4 -edge-connected graph admits a 3 -flow. Proof
3-flows in arc-transitive Tutte’s 3-flow conjecture graphs Circulations and integer flows Tutte’s flow conjectures Conjecture 3-flows in vertex- (The 3 -flow conjecture) transitive graphs Every 4 -edge-connected graph admits a 3 -flow. Proof Theorem (M. Kochol 2001) The 3 -flow conjecture is true if and only if every 5 -edge-connected graph admits a 3 -flow.
3-flows in arc-transitive recent breakthrough graphs Circulations and integer flows Tutte’s flow conjectures 3-flows in Theorem vertex- transitive (C. Thomassen 2012) graphs Every 8 -edge-connected graph admits a 3 -flow. Proof
3-flows in arc-transitive recent breakthrough graphs Circulations and integer flows Tutte’s flow conjectures 3-flows in Theorem vertex- transitive (C. Thomassen 2012) graphs Every 8 -edge-connected graph admits a 3 -flow. Proof Theorem (L. M. Lov´ asz, C. Thomassen, Y. Wu and C. Q. Zhang 2013) Every 6 -edge-connected graph admits a 3 -flow.
3-flows in arc-transitive motivation graphs Circulations and integer flows Tutte’s flow Theorem conjectures (M. E. Watkins 1969; W. Mader 1970) 3-flows in vertex- Every vertex-transitive graph of valency d is d-edge-connected. transitive graphs Proof
3-flows in arc-transitive motivation graphs Circulations and integer flows Tutte’s flow Theorem conjectures (M. E. Watkins 1969; W. Mader 1970) 3-flows in vertex- Every vertex-transitive graph of valency d is d-edge-connected. transitive graphs Proof Conjecture (Vertex-transitive version of the 3 -flow conjecture) Every vertex-transitive graph of valency at least 4 admits a 3 -flow. It suffices to prove this for vertex-transitive graphs of valency 5.
3-flows in arc-transitive 3-flows in Cayley graphs on graphs nilpotent groups Circulations and integer flows Tutte’s flow Theorem conjectures 3-flows in (P. Potaˇ cnik 2005) vertex- transitive Every Cayley graph of valency at least 4 on a finite abelian graphs group admits a 3 -flow. Proof
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