Paths in Graphs and Continua Paul Gartside May 2018 University of Pittsburgh
Joint work with: Max Pitz, University of Hamburg and Benjamin Espinoza, University of Pittsburgh - Greensburg Ana Mamatelashvili, Melbourne
Building and Crossing Bridges Problem (Euler) For which graphs G is there a path crossing every edge exactly once? 3
Building and Crossing Bridges Problem (Euler) For which graphs G is there a path crossing every edge exactly once? 3
What is a Graph? A Path? ‘Graph’ G : Combinatoric object: vertices V , edges E . . . Topological object: 1-complex. ‘Path’ (from a to b ): combinatorially or topologically. 4
Euler’s Solution Theorem (Euler) Let G = ( V , E ) be a finite graph. Then TFAE: (a) there is a closed path in G crossing each edge exactly once (b) every vertex has even degree, and (c) for every partition, A , B of V the number of edges starting in A and ending in B is even. 5
Euler’s Solution Theorem (Euler) Let G = ( V , E ) be a finite graph. Then TFAE: (a) there is an open path in G crossing each edge exactly once (b) every vertex except 2 has even degree, and (c) there are 2 vertices a , b such that for every partition, A , B of V with a in A and b in B the number of edges starting in A and ending in B is even. 5
Hamilton’s Problem Problem (Hamilton) For which graphs G is there a path visiting every vertex exactly once? Note: a Hamiltonian open path is, topologically, an arc ( ∼ = [0 , 1]). There is no characterization of Hamiltonian graphs. 6
Steps in The Plan Hamilton Euler Graphs Graph-like Continua 7
Citizens! Assert Your Rights! Note: G = ( V , E ) open Hamiltonian iff the vertices V are contained in an arc. Definition A graph G is n -open Hamilton iff any points x 1 , . . . , x n are contained in an arc. Definition A graph G is n -closed Hamilton iff any points x 1 , . . . , x n are contained in a circle. 8
Citizens! Assert Your Rights! Note: G = ( V , E ) open Hamiltonian iff the vertices V are contained in an arc. Definition A space X is n -arc connected iff any points x 1 , . . . , x n are contained in an arc. Definition A space X is n -circle connected iff any points x 1 , . . . , x n are contained in a circle. 8
They will (not) control us Theorem A graph G is 6 -ac if and only if either G is 7 -ac or, after suppressing all degree-2-vertices, the graph G is 3 -regular, 3 -connected, and removing any 6 edges does not disconnect G into 4 or more components. 9
7 = ∞ Theorem Let G be a non-degenerate graph. Then the following are equivalent: (a) G is 7 -ac, (b) G is n-ac, for all n, and (c) after suppressing or adding degree 2 vertices G is isomorphic to one of 9 graphs. 10
Menger Theorem (Menger’s Theorem) Let G = ( V , E ) be a (potentially infinite) graph and A , B ⊆ V . Then the minimum number of vertices separating A from B in G is equal to the maximum number of disjoint A − B paths in G. But we need to use algorithmic versions using alternating paths. 11
We have. . . Characterized n -ac and n -cc graphs G for: all n , and all graphs. 12
From Finite to Infinite Graphs Problem How to lift results from finite graphs to infinite graphs. 13
From Finite to Infinite Graphs Problem How to lift results from finite graphs to countable, locally finite graphs. 13
Eulerianity Theorem (Euler) Let G be a finite (connected) graph. Then: G is Eulerian (there is a closed path crossing every edge exactly once) if and only if every vertex is even. 14
Eulerianity Theorem (Euler) Let G be a finite (connected) graph. Then: G is open Eulerian (there is an open path crossing every edge exactly once) if and only if two vertices are odd, the rest even. 14
Problems with Infinite Graphs 15
4 -Regular Tree Sabidussi: connected graph is Eulerian if every vertex even and just one end. 16
Solution: Add the Ends, and Compactify Diestel. Berger, Bowler, Bruhn, Carmesin, Christian, Georgakopoulos, Richter, Rooney, Stein. R. Diestel, Locally finite graphs with ends: a topological approach I-III , Discrete Math (2010–11). 17
Freudenthal Compactification Adding all ends gives the Freudenthal compactification. Equivalently, maximal compactification with 0-dimensional remainder. Finite graph G Freudenthal compactification γ G cycle circle path ‘standard’ path closed path ‘standard’ loop edge-disjoint edge-disjoint Eulerian Eulerian 18
Graph-Like Spaces and Continua Definition (Thomassen and Vella, 2008) A graph-like space (respectively, continuum) is a triple ( X , V , E ) where: X is a compact, metrizable space (respectively, continuum), V ⊆ X is a closed zero-dimensional subset, and E is a discrete index set such that X \ V ∼ = E × (0 , 1). The Freudenthal compactification of a locally finite graph is graph-like. 19
Cantor Bouquet of Circles I CBC 20
Characterizing Graph-Like Continua Theorem The following are equivalent for a continuum X: (i) X is graph-like, (ii) X is completely regular, (iii) X is a countable inverse limit of finite connected multi-graphs with onto, monotone bonding maps that project vertices onto vertices, and (iv) X is homeomorphic to a connected standard subspace of a Freudenthal compactification of a locally finite graph. 21
Eulerian Graph-like Continua Theorem Let X be a graph-like continuum with vertices V . TFAE: (i) X is closed [open] Eulerian, (ii) every vertex is even [apart from precisely two vertices which are odd], and (iii) [there are vertices x � = y such that] for every partition of V into two clopen pieces, the number of cross edges is even [if and only if x and y lie in the same part]. 22
Back (Briefly) to the Hamilton Side Theorem For the Freudenthal compactification γ G of a locally finite connected graph G and for each n ≥ 2 : γ G is n-ac if and only if G is n-ac. Theorem For every n ≥ 2 , there are 2 ℵ 0 many non-homeomorphic graph-like continua which are n-ac [n-cc] but not ( n + 1) -ac [ ( n + 1) -cc]. 23
Eulerian Continua A continuum is Eulerian if it satisfies any of following conditions: Theorem For a continuous surjection f : S 1 → X onto X, TFAE: (1) f is arcwise increasing; (2) f is irreducible. (3) f is hereditarily irreducible; (4) f is strongly irreducible; (5) f is almost injective; and Moreover, if X has a dense collection of free arcs E, then also (6) f traverses every edge exactly once, as an embedding, and f − 1 ( E ) dense in S 1 . 24
Our Conjecture Gartside, Pitz A Peano continuum X is Eulerian if and only if X ∼ is Eulerian, where X ∼ is the graph-like continuum obtained by identifying to points all components of � X \ { all open free arcs } . This strengthens conjecture of Bula, Nikiel and Tymchatyn. 25
Alternative Phrasing Gartside, Pitz Let X be a Peano continuum X . Write E = � { all open free arcs } and V = X \ E . Then X is Eulerian if and only if for every partition of V induced by A , B open in X the number of edges (components of E ) from A to B is even. “All edge-cuts even” 26
Two Reductions Theorem (Espinoza, Matsuhashi) Continua without free arcs are Eulerian. So we need only consider continua with free arcs. And wlog these arcs are dense: Theorem Let X be a Peano continuum with free arcs indexed by E, and let D be a countable dense subset for F = X \ E. Let X ′ = X ∪ L be the Peano continuum where we attach a zero-sequence of loops L = { ℓ d } : d ∈ D to points in D. Then if X ′ is Eulerian, then so is X. 27
Eulerianity conjecture for Peano graphs Conjecture (Gartside & Pitz): A Peano graph is Eulerian if and only if all its edge-cuts are even. Peano graph: Peano compactification of locally finite, countable graph G via boundary ∂ G . Conjecture true – when ∂ G = S 1 . 28
Framework: Approximation by finite Eulerian graphs 29
Framework: Approximation by finite Eulerian graphs 1 Partition into almost Eulerian tiles. (This step uses Bing’s Brick Partition Theorem and the theory of TST’s, fundamental circuits and thin sums by Diestel et al...). 30
Framework: Approximation by finite Eulerian graphs G 1 1 Partition into almost Eulerian tiles. (This step uses Bing’s Brick Partition Theorem and the theory of TST’s, fundamental circuits and thin sums by Diestel et al...). 2 Let G 1 be graph on the tiles with edge set all uncovered edges. 30
Framework: Approximation by finite Eulerian graphs G 1 3 Carefully add dummy edges to G 1 in order to make it Eulerian. (This step uses the assumption that all cuts of X are even). 30
Framework: Approximation by finite Eulerian graphs G 1 3 Carefully add dummy edges to G 1 in order to make it Eulerian. 4 Add one dummy loop for each new dummy edge at the intersection of corresponding tiles. 30
Framework: Approximation by finite Eulerian graphs G 1 3 Carefully add dummy edges to G 1 in order to make it Eulerian. 4 Add one dummy loop for each new dummy edge at the intersection of corresponding tiles. 5 Repeat! 30
Framework: Approximation by finite Eulerian graphs G 1 1 ′ Partition each tile into (smaller) almost Eulerian tiles. 30
Framework: Approximation by finite Eulerian graphs G 2 2 ′ Obtain a “finer” graph G 2 on the new tiles. 31
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