Infinite circuits in infinite graphs
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1 Infinite circuits in infinite graphs Henning Bruhn Universität Hamburg R. Diestel, A. Georgakopoulos, D. Kühn, P. Sprüssel, M. Stein Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 1 / 25
2 Locally Finite Graphs (Almost) every graph in this talk will be locally finite Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 2 / 25
3 Infinite circuits The Cycle Space of a Finite Graph Cycle Space C(G) for finite G: set of all symmetric differences of circuits Z 2 -vector space 1. homology group Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 3 / 25
4 Infinite circuits The Cycle Space of a Finite Graph Cycle Space C(G) for finite G: set of all symmetric differences of circuits Z 2 -vector space 1. homology group Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 3 / 25
5 Infinite circuits The Cycle Space of a Finite Graph Cycle Space C(G) for finite G: set of all symmetric differences of circuits Z 2 -vector space 1. homology group Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 3 / 25
6 Infinite circuits MacLane s Planarity Criterion Theorem (MacLane) If G is finite then G is planar iff C(G) has simple generating set. simple: no edge in two members face boundaries generate C(G) Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 4 / 25
7 Infinite circuits MacLane Fails in Infinite Graphs C need infinite sums Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 5 / 25
8 Infinite circuits Too few circuits e face boundaries containing e infinite cannot generate circuits containing e need more circuits Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 6 / 25
9 Infinite circuits Too few circuits e face boundaries containing e infinite cannot generate circuits containing e need more circuits Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 6 / 25
10 Infinite circuits Infinite Face Boundaries Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 7 / 25
11 The Monster Circuit Infinite circuits D D l D D e l e l 1 combinatorial description seems hopeless Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 8 / 25
12 Infinite circuits Topology on G+ends Define topology on G+ends ~ 0 1 S C ω on G: topology of 1-complex Freudenthal compactification Hausdorff Henning Bruhn (U Hamburg) Infinite circuits Haifa 08 9 / 25
13 Circles Infinite circuits For locally finite G Circle homeomorphic image of S 1 in G+ends Circuit edge set of a circle Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
14 Infinite circuits Infinite circuits: examples no cycle: Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
15 Infinite circuits Topological Cycle Space Topological cycle space C(G) set of thin sums of circuits (Diestel&Kühn) we allow infinite sums! C(G) 1. homology group Diestel&Sprüssel Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
16 Infinite circuits Topological Cycle Space Topological cycle space C(G) set of thin sums of circuits (Diestel&Kühn) we allow infinite sums! C(G) 1. homology group Diestel&Sprüssel Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
17 Infinite circuits MacLane s Planarity Criterion Theorem (MacLane) If G is finite then G is planar iff C(G) has simple generating set. Theorem (Bruhn&Stein) If G is locally finite then G is planar iff C(G) has simple generating set. e Prior work due to Thomassen ( 80), also Bonnington&Richter ( 03) Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
18 Infinite circuits C(G) is the Right Concept A host of results: orthogonality of circuits and cuts [Diestel & Kühn] cycle space elements are disjoint union of circuits [DK] Tutte s generating theorem [Bruhn] Tutte s planarity criterion [B&Stein] Gallai s cycle-cocycle partition [BDS] Whitney s planarity criterion [BD] duality of spanning trees [BD] characterisation by degrees [BS] tree packing [S] Fleischner s theorem [Georgakopoulos] C(G) is generated by geodesic circuits [G&Sprüssel] Also: work by Richter&Vella in more general context Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
19 Tutte s planarity criterion Tutte s Planarity Criterion Peripheral circuit: non-separating and without chords Theorem (Tutte) If G is finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. Theorem (Bruhn&Stein) If G is locally finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
20 Tutte s planarity criterion Tutte s Planarity Criterion Peripheral circuit: non-separating and without chords Theorem (Tutte) If G is finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. e Theorem (Bruhn&Stein) If G is locally finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
21 Tutte s planarity criterion Tutte s Planarity Criterion Peripheral circuit: non-separating and without chords Theorem (Tutte) If G is finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. e Theorem (Bruhn&Stein) If G is locally finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
22 Tutte s planarity criterion Tutte s Planarity Criterion Peripheral circuit: non-separating and without chords Theorem (Tutte) If G is finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. e Theorem (Bruhn&Stein) If G is locally finite and 3-connected then G is planar iff every edge lies in at most two peripheral circuits. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
23 Gallai s edge partition Gallai s edge partition a cut Theorem (Gallai) For finite G, there is a partition Z F = E(G), so that F is a cut and Z C(G). Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
24 Gallai s edge partition Gallai s edge partition unique partition need infinite circuits Theorem (Bruhn, Diestel & Stein) For locally finite G, there is a partition Z F = E(G), so that F is a cut and Z C(G). Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
25 Gallai s edge partition Gallai s edge partition unique partition need infinite circuits Theorem (Bruhn, Diestel & Stein) For locally finite G, there is a partition Z F = E(G), so that F is a cut and Z C(G). Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
26 Hamilton circuits Hamilton Circuits in Finite Graphs Theorem (Tutte) If G is finite planar and 4-connected then it has a Hamilton circuit. Conjecture (Nash-Williams) If G is locally finite planar and 4-connected with at most two ends then it has a spanning double ray. Yu has announced a proof of NW s conjecture Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
27 Hamilton circuits Infinite Hamilton Circuits Hamilton circuit: spanning (infinite) circuit can go through arbitrarily many ends Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
28 Hamilton circuits Hamilton Circuits in Planar Graphs Conjecture If G is locally finite planar and 4-connected then it has an infinite Hamilton circuit. Two preliminary results Theorem (Bruhn&Yu) If G is locally finite, planar, 6-connected and has only finitely many ends then it has a Hamilton circuit. Theorem (Cui, Wang & Yu) If G is locally finite, planar, 4-connected and has a VAP-free drawing then it has a Hamilton circuit. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
29 Hamilton circuits Fleischner s Theorem Square of G: put in an edge between any two vertices of distance 2 Theorem (Fleischner) The square of a 2-connected finite graph has a Hamilton circuit. Theorem (Georgakopoulos) The square of a 2-connected locally finite graph has a Hamilton circuit. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
30 Hamilton circuits Fleischner s Theorem Square of G: put in an edge between any two vertices of distance 2 Theorem (Fleischner) The square of a 2-connected finite graph has a Hamilton circuit. Theorem (Georgakopoulos) The square of a 2-connected locally finite graph has a Hamilton circuit. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
31 Methods Methods are combinatorial connected & locally finite countable cut criterion Theorem (Diestel & Kühn) G locally finite. Then Z C(G) iff Z F =even for every finite cut F. Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
32 Limits of circuits Methods limit of circuits C 1, C 2,... limit of circuits C(G) construction from local to global possible Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
33 Limits of circuits Methods limit of circuits C 1, C 2,... limit of circuits C(G) construction from local to global possible Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
34 Limits of circuits Methods limit of circuits C 1, C 2,... limit of circuits C(G) construction from local to global possible Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
35 Limits of circuits Methods limit of circuits C 1, C 2,... limit of circuits C(G) construction from local to global possible Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
36 Limits of circuits Methods limit of circuits C 1, C 2,... limit of circuits C(G) construction from local to global possible Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
37 Methods Compactness arguments Theorem (Kőnig s Infinity Lemma) Let W 1, W 2,... be finite, non-empty and disjoint, let H graph with V (H) = W n. If each vertex in W n+1 has neighbour in W n then there is a ray w 1 w 2... with w n W n for all n. standard tool uncountable Tychonov W 1 W 2 W 3 W 4 W 5 Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
38 Methods Compactness arguments Theorem (Kőnig s Infinity Lemma) Let W 1, W 2,... be finite, non-empty and disjoint, let H graph with V (H) = W n. If each vertex in W n+1 has neighbour in W n then there is a ray w 1 w 2... with w n W n for all n. standard tool uncountable Tychonov W 1 W 2 W 3 W 4 W 5 Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
39 Methods Application of Infinity Lemma assume: can colour finite subgraphs colouring for whole graph usually application not this straightforward works only in easy cases Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
40 Methods Application of Infinity Lemma assume: can colour finite subgraphs colouring for whole graph usually application not this straightforward works only in easy cases Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
41 Methods Application of Infinity Lemma assume: can colour finite subgraphs colouring for whole graph usually application not this straightforward works only in easy cases Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
42 Methods Application of Infinity Lemma assume: can colour finite subgraphs colouring for whole graph usually application not this straightforward works only in easy cases Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
43 Methods Application of Infinity Lemma assume: can colour finite subgraphs colouring for whole graph usually application not this straightforward works only in easy cases Henning Bruhn (U Hamburg) Infinite circuits Haifa / 25
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