On the Hamiltonicity of Line Graphs of Locally Finite, 6-Edge-Connected Graphs Richard C. Brewster1 and Daryl Funk2 1 DEPARTMENT

OF MATHEMATICS AND STATISTICS THOMPSON RIVERS UNIVERSITY KAMLOOPS, BC, CANADA E-mail: [email protected] 2 DEPARTMENT

OF MATHEMATICS

SIMON FRASER UNIVERSITY BURNABY, BC, CANADA E-mail: [email protected]

Received September 17, 2010; Revised August 2, 2011 Published online 22 November 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jgt.20641

Abstract: The topological approach to the study of infinite graphs of Diestel and K ¨ hn has enabled several results on Hamilton cycles in finite graphs to be extended to locally finite graphs. We consider the result that the line graph of a finite 4-edge-connected graph is hamiltonian. We prove a weaker version of this result for infinite graphs: The line graph of locally finite, 6-edge-connected graph with a finite number of ends, each of which is thin, is hamiltonian. 䉷 2011 Wiley Periodicals, Inc. J Graph Theory 71: 182--191, 2012 Keywords: locally finite; Hamilton circle; line graph

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INTRODUCTION

The question of how one should extend the concept of a Hamilton cycle to infinite graphs has received much attention in the literature. Studies of the cycle space of infinite graphs include those of Bonnington and Richter [1], Casteels and Richter [8], Diestel and K¨uhn [13–15], and Vella and Richter [25]. We follow the approach of Diestel and K¨uhn, which has yielded extensions of several well-known results on cycles in finite graphs to locally finite graphs (for instance, see [2–6, 20, 19]). In this context, a Hamilton circle in a graph G is a homeomorph of the unit circle in the Freudenthal compactification |G| of G containing all vertices and ends of G. Bruhn and Yu [7] have partially extended Tutte’s result that all finite 4-connected planar graphs have a Hamilton cycle [24] by showing that all 6-connected planar locally finite graphs with only finitely many ends contain a Hamilton circle. Bruhn has conjectured (see [10]) that this holds for locally finite 4-connected planar graphs. Cui et al. have verified the conjecture, for graphs admitting a drawing in the plane which has no vertex accumulation point [9]. A classic theorem of Fleischner, that the square of a 2-connected finite graph has a Hamilton cycle [16], has been extended by Georgakopoulos to locally finite graphs [18]. This result had previously been extended to locally finite graphs with just one end by Thomassen [22]. Thomassen has conjectured that every finite 4-connected line graph is hamiltonian. As mentioned above, it is known that the line graph of a 4-edge-connected graph is hamiltonian [23]. Georgakopoulos has conjectured this result holds for locally finite graphs [17]. In this article, we present a partial result toward this conjecture. Theorem 1. Let G be a 6-edge-connected, locally finite graph. Further suppose G has a finite number of ends, each of which is thin. Then the line graph L(G) is hamiltonian.

2.

DEFINITIONS AND PREVIOUS RESULTS

A.

Topological Concepts

Let G = (V, E) be a locally finite graph with vertex set V and edge set E. A ray is a one-way infinite path. The subrays of a ray are its tails. An end of G is an equivalence class of rays, where two rays R1 and R2 are equivalent if for any finite S ⊂ V(G), R1 and R2 contain a tail in the same component of G−S. We often make use of the fact that two rays are equivalent if, and only if, there is an infinite collection of pairwise disjoint paths linking the rays. The vertex-degree of an end is the maximum number of disjoint rays in the end. An end is thin if it has finite vertex-degree. The set of ends of G is denoted  = (G). The associated topological space, obtained by taking the Freudenthal compactification of G viewed as a 1-complex, is denoted |G|. For an edge e = uv ∈ E(G), with u, v ∈ V(G), the set of inner points of e ⊂ |G| is the set of points e˚ := e\{u, v}. A circle in |G| is a homeomorphic image of the unit circle S1 ⊂ R2 . The set of edges contained in a circle is called its circuit. A family {Ai }i∈Iof subsets of E is called thin if no edge lies in Ai for infinitely many i. The sum i∈I Ai of a thin family is the set of all edges which are in an odd number of subsets Ai , i.e. edges are added modulo 2. The (topological) cycle space of Journal of Graph Theory DOI 10.1002/jgt

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a locally finite graph G is the subspace of the edge space of G consisting of all sums of thin families of circuits, denoted C (G). Lemma 2 (Diestel [11], Corollary 8.5.9). nite (thin) sums.

The cycle space C (G) is closed under infi-

For details on |G| and its cycle space C (G), see Chapter 8.5 of [11]. Surveys are provided by Diestel [10] and [12]. Our proof of Theorem 1 is based on a proof for the finite case employing disjoint spanning trees. We review some key definitions and results for spanning trees in infinite graphs. A topological spanning tree of G is an arc-connected standard subspace of |G| which contains all of V(G)∪(G), but contains no circle. Lemma 3 (Diestel [11], Corollary of Theorem 8.5.10). Every 2k-edge-connected locally finite multigraph G has k edge-disjoint topological spanning trees. Analogous to finite graphs, with respect to addition or deletion of edges, a topological spanning tree T is minimally arc-connected and maximally contains no circle. For every e ∈ E not in T, T ∪e contains a unique circle whose edges form the fundamental circuit (T, e) of e with respect to T. For every edge e ⊆ T, the subspace T \ e˚ has exactly two arc-components. The set of edges between them is the fundamental cut (T, e) of G with respect to T and e. Lemma 4 (Diestel and K¨uhn [15], Lemma 6.2). The fundamental circuits of a locally finite graph with respect to any topological spanning tree form a thin family. A proof that the line graph of a finite 4-edge-connected graph G is hamiltonian can be outlined as follows: In G there are two edge-disjoint spanning trees T1 and T2 . The  sum e∈T1 (T2 , e) is a spanning eulerian subgraph of G, from which a Hamilton cycle in L(G) can be constructed. We proceed along the similar lines, and thus require some knowledge of eulerian subgraphs in the infinite setting. For a graph G, a closed trail in the space |G| is a continuous map : S1 → |G| which is injective on E˚ ⊂ |G| (the set of inner points of edges of G). A closed dominating trail in |G| is a closed trail which contains at least one endvertex of every edge of G. A topological Euler tour of |G| is a closed trail in |G| which traverses every edge of G. Theorem 5 (Georgakopoulos [19], Theorem 1.3). Let H ⊆ G be locally finite multigraphs such that the closure H of H in |G| is topologically connected. Then E(H) ∈ C (G) if and only if H admits a topological Euler tour in |G|. A key step in our proof is to ensure that the spanning eulerian subgraph visits each end of |G| only once. The existence of such a tour follows from a theorem of Georgakopoulos. Theorem 6 (Georgakopoulos [18], Theorem 4). If a locally finite multigraph G has a topological Euler tour, then it also has one that is injective on the set of ends (G) ⊂ |G| of G. The caveat in using the above theorem is that we wish to apply the theorem to a spanning eulerian subgraph H ⊆ G. The tour returned by the theorem is injective on the Journal of Graph Theory DOI 10.1002/jgt

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ends of the H which may not have the same end topology as the host supergraph G. Hence, we require the following concept. Definition (Section 7 of [15]). A subgraph H ⊆ G is end-faithful if the the cannonical projection  : (H) → (G) taking every end of H to the end of G which contains it (as a subset of rays) is a bijection. The crux of our proof is to modify a spanning eulerian subgraph H in G so that H is end-faithful. Lemma 7 (Georgakopoulos [18], Lemma 6). Suppose H is a spanning end-faithful subgraph of G. Then the cannonical projection  : (H) → (G) is a homeomorphism. Hence if H ⊆ G is a spanning end-faithful subgraph, we may identify the ends  ∈ (H) and () ∈ (G), say H and G have the same set of ends, and write (H) = (G). The following lemma is crucial throughout our work. A comb is the union of a ray R with infinitely many disjoint finite paths all with their initial vertex on R; R is the spine; the last vertices of those paths are called teeth. Lemma 8 (Star-Comb Lemma [11], Lemma 8.2.2). Let U be any infinite set of vertices in a connected graph G. Then G either contains a comb with all teeth in U or a subdivision of an infinite star with all leaves in U. When G is locally finite, applying the Star-Comb Lemma to any infinite set of vertices always guarantees the existence of a comb. While we write G for the graph and |G| for its associated topological space, we write T for both T ⊆ |G| a topological spanning tree, an arc-connected standard subspace of ˚ |G|, and for the subgraph of G induced by T ∩{V(G)∪ E(G)} (or T \(G)). A topological spanning tree T is arc-connected, but as a subgraph, T may not be graph theoretically connected. Given these two notions of connectedness, for the sake of clarity, we always specify ‘topological component’ or ‘graph theoretic component’ in these discussions.

B.

A Theorem of Harary & Nash-Williams, for Locally Finite |G|

The following is a classic theorem of Harary and Nash-Williams. Theorem 9 (Harary and Nash-Williams [21]). Let G be a finite graph with at least four vertices. Then the line graph L(G) is hamiltonian if, and only if, G has a closed trail which includes at least one endvertex of each edge of G or G is isomorphic to K1,s for some integer s ≥ 3. We first extend the above theorem to the space |G| for locally finite graphs. This extension is a strengthening of Corollary 12 in [18] in which a topological Euler tour in |G| is used to established L(G) is hamiltonian. Lemma 10. Let G be a locally finite graph. If |G| contains a closed dominating trail which is injective on (G), then its line graph L(G) is hamiltonian. Proof. The proof follows that of Corollary 12 in [18]. Let  : S1 → |G| be a closed dominating trail in |G| which is injective on (G). Denote F = (S1 ). Journal of Graph Theory DOI 10.1002/jgt

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Define an embedding  : S1 → |L(G)| as follows. Let u0 , u1 , u2 , . . . be an enumeration of V(F). • Suppose e ∈ E(F). Since  is continuous and injective on e, −1 (˚e) is an open interval in S1 . Let me be its midpoint. Set  (me ) = e ∈ V(L(G)). • For each interval I of S1 mapped by  to a trail xeye z define  to map the open interval with endpoints me , me continuously and bijectively to the edge ee ∈ E(L(G)). Now  is defined on −1 (V(F)∪E(F)). Since  is dominating, all edges not in F are incident with a vertex of F. We now modify  to include these edges in its image. • For each ui , i = 0, 1, 2, . . ., there are edges e and e incident with ui in G such that  maps the interval (me , me ) to the edge ee in L(G). Modify  to map the interval to the finite path e1 e2 · · ·ek in L(G) where this path is maximum subject to the following constraints: ◦ ◦ ◦ ◦

e1 = e, ek = e , each of e1 , e2 , . . . , ek are edges in G incident with ui , e2 , . . . , ek−1 ∈/ E(F), and for  = 2, . . . , k −1, if e = ui v, then v = uj for j
Finally, given a ray R in  ∈ (L(G)), the vertices of R correspond to a sequence of adjacent edges in G in which one can find a ray R belonging to an end, say  ∈ (G). Let the vertices of R be v0 , v1 , v2 , . . . Since F is a dominating trail, for each edge of R, say vi vi+1 , either vi or vi+1 must belong to F. The sequence of vertices v0 , v1 , v2 , . . . converges to . The subsequence of these vertices that belong to F must also converge to , as (S1 ) = F is compact. In particular,  ∈ (S1 ); moreover, by injectivity, there is only one point of S1 mapped to  by . Define  (−1 ()) =  . As noted in [18], (G) and (L(G)) have the same topology. (In a locally finite graph, each finite set of vertices is incident with a finite number of edges, and each finite set of edges covers a finite number of vertices.) Thus  is a continuous and injective map from a compact space to Hausdorff space. Hence,  is a homeomorphism to its image which contains all of V(L(G)) and (L(G)). 

3.

CONSTRUCTION OF A DOMINATING TRAIL

A.

Constructing a Spanning Eulerian Subgraph F ⊆ G

Let G = (V, E) be a locally finite 6-edge-connected graph with a finite number of ends, each of which is thin. We begin by constructing a spanning eulerian subgraph of G which may or may not be end-faithful. By Lemma 3, |G| contains three edge-disjoint topological spanning trees, T1 , T2 , T3 . By Lemma 4, the fundamental circuits of T2 with respect to T1 form a thin family. Hence, the following sum is well defined. Define F to be the topological closure of the spanning subgraph of G with edge set; E(F) =

 e∈T1

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(T2 , e).

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Lemma 11.

The graph F is a spanning eulerian subgraph of G.

Proof. Since E(T1 ) ⊆ F implies T1 = E(T1 ) ⊆ F, F is spanning. To see that F is topologically connected, pick any two points x, y ∈ F. If x is an inner point of an edge in E(F), then there is an arc in F linking x to a vertex. Hence without loss of generality, we may assume x, y ∈ V ∪. There is an arc in T1 ⊆ F linking x and y. By Lemmas 2 and 4, F is an element of the cycle space of G. By Theorem 5, F is eulerian. 

B.

Constructing an End-Faithful Subgraph F ∗ ⊇ F

In this section, we construct F ∗ ⊇ F such that F ∗ is an end-faithful eulerian subgraph of G. If F is already end-faithful, then define F ∗ = F. Otherwise, proceed as below. We first examine under what conditions F is not end-faithful. We begin by looking at the graph theoretic versus topological connectivity of the spanning trees. Lemma 12.

If T1 is connected as a subgraph, then F is end-faithful.

Proof. Let  : (F) → (G) be the projection map taking each end of F to the end of G which contains it. (Surjectivity) Let  ∈ (G). By the Star-Comb Lemma, there is a ray R contained in T1 in . Since T1 ⊆ F, R ⊆ F. Let  ∈ (F) be the end of F containing R. Then ( ) = . (Injectivity) Given distinct ends  1 ,  2 ∈ (F), the Star-Comb Lemma ensures that T1 contains disjoint rays R1 ∈  1 and R2 ∈  2 . Since T1 is connected as a subgraph, there is path in T1 from R1 to R2 . Consequently, R1 and R2 must belong to different ends of G, as T1 is circle free. In particular, ( 1 ) = ( 2 ).  Since F is not end-faithful, by Lemma 12 T1 has multiple graph theoretic components. The strategy is to add edges between graph theoretic components of T1 with rays belonging to a common end in G. The result is an end-faithful spanning eulerian subgraph. Specifically, from F we construct F ∗ as follows. For each n ∈ N, we construct an eulerian spanning subgraph Fn , with E(T1 ) ⊆ E(Fn ). A nested sequence of subsets of vertices S0 ⊂ S1 ⊂ S2 ⊂ · · · is constructed so that edges added at step n are in G−Sn , and belong to G[Sn+1 ]. In the limit lim infFn =

∞  ∞ 

Fi = F ∗

n=1 i=n

has the same end topology as G. Set F0 = F. Lemma 13. If G is a locally finite graph with finitely many ends, each of which is thin, then every topological spanning tree of G has only finitely many graph theoretic components. Proof. Let T be a topological spanning tree of G. Since T is arc-connected, every graph theoretic component of T is infinite (for instance, see Lemma 8.5.5 of [11]). By the Star-Comb Lemma, each graph theoretic component of T contains a ray in some end of G. By assumption, G has a finite number of ends, each of which is thin. Hence, T has a finite number of graph theoretic components.  Journal of Graph Theory DOI 10.1002/jgt

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It is easy to see that for a locally finite graph deleting finitely many vertices never changes the number of components from finite to infinite. Hence, Lemma 13 implies that for any finite S ⊆ V(G), Ti −S (for i = 1, 2, 3) has a finite number of graph theoretic components. Let v0 , v1 , . . . be an enumeration of V(G). Let S0 = {v0 , v1 , . . . , vk } be large enough so that each graph theoretic component of T1 −S0 has a ray in at most one end of G. Since (G) is finite, such a set S0 exists. In the following, let X1 , X2 , . . . , X be the graph theoretic components of T1 −S0 with a ray in a common end, say . An edge with one end in Xi and the other in Xj for some i = j is called a crossing edge. Lemma 14. Let S ⊆ V(G) be a finite set. Suppose there are an infinite number of crossing edges between Xi and Xj . Then there is a crossing edge e from Xi to Xj such that the fundamental circuit of e with respect to Ti (i = 2 or 3) is wholly contained in G−S. Proof. The crossing edges between Xi and Xj can be partitioned into three sets: those edges in T2 , say A; those edges in T3 , say B; and those edges in neither, say C. Either A∪C or B∪C is infinite. Without loss of generality, assume the former. Since there are only a finite number of edges in G[S], the subgraph of G induced by S, and the set of fundamental circuits of G with respect to T3 is thin, there must be an edge e ∈ A∪C with (T3 , e) ⊆ G−S.  To construct Fn+1 from Fn proceed as follows. Consider, as above, an end  and the components X1 , X2 , . . . , X with a ray in T1 −S0 belonging to . Construct an auxilarly graph H = H(). Let V(H) = {x1, x2 , . . . , x }. For each pair of vertices, add the edge xi xj if, and only if, there are an infinite number of crossing edges between Xi and Xj . For each edge xi xj in H, consider the crossing edges between Xi and Xj belonging to G−Sn . If some crossing edge e = vr vs already belongs to Fn , simply enlarge Sn to include both endvertices of e, i.e. set Sn = {v0, v1 , . . . , vmax{r,s} }. Otherwise, by Lemma 14 there is a crossing edge e in G−Sn , and a fundamental circuit C = (Tt , e), where t = 2 or 3, such that C is wholly contained in G−Sn . Add C (edges are added modulo 2) to Fn . Again, enlarge Sn to include both endvertices of e. Repeat this for each edge in H. Similarly, repeat these crossing edge additions for each end  in G with the appropriate auxilarly graph H( ). Call the resulting eulerian subgraph Fn+1 . The vertex set Sn has been enlarged to include all of the added crossing edges’ endvertices. Call the resulting set Sn+1 . Lemma 15.

The graph H is connected.

Proof. Suppose to the contrary there is a subset of vertices U of H such that there are no edges in the cut E(U, V(H)−U). Then for each vertex in xi ∈ U and for each vertex xj ∈ V(H)−U there is a finite number of crossing edges between Xi and Xj . Hence for sufficiently large S, there are no crossing edges between Xi −S and Xj −S. Since both U and V(H)−U are finite, we can in fact choose S large enough such there are no edges from any component Xr , xr ∈ U to any component Xs , xs ∈ V(H)−U. Consequently, a ray in Xi and a ray in Xj can be separated by a finite set S, contrary to the fact they both belong to .  Journal of Graph Theory DOI 10.1002/jgt

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Lemma 16. There are an infinite number of vertex disjoint finite paths between Xi and Xj in F ∗ . Proof. By Lemma 15, there is a path xi = u0 u1 u2 . . . uk = xj in H. Let U0 , U1 , . . . , Uk be the corresponding components in T1 −S0 . For any S ⊆ V(G), the subgraphs U0 −S, U1 −S, . . . , Uk −S each have exactly one infinite component. Call these infinite components respectively U0 , U1 , . . . , Uk . Because a crossing edge, once added to Fn , is never removed by the choices of subsequent crossing edges and fundamental cycles, the crossing edge belongs to F ∗ . Hence in F ∗ there is a crossing edge from U0 to U1 , say w0 w1 , belonging to F ∗ −S. Similarly, there is a crossing edge from U1 to U2 , say w2 w3 belonging to F ∗ −S. Observe that no edge of a fundamental cycle added to Fn to construct Fn+1 is contained in T1 . Thus, no edge of T1 is ever removed from any Fn . Since T1 ⊆ F, it follows T1 ⊆ F ∗ . Since U1 is a graph theoretic component, there is also a path from w1 to w2 in U1 ⊆ T1 −S ⊆ F ∗ −S. Hence, there is a path from w0 to w3 in F ∗ −S. Continuing in this manner, we construct a path from Xi to Xj in the subgraph F ∗ −S.  Lemma 17.

The graph F ∗ is end-faithful.

Proof. Let  : (F ∗ ) → (G) be the projection map taking each end of F ∗ to the end of G which contains it. Since T1 ⊆ F ∗ , clearly  is surjective (as in the proof of Lemma 12). Now suppose (1 ) = (2 ) = , for some 1 , 2 ∈ (F ∗ ),  ∈ (G). Let R1 ∈ 1 and R2 ∈ 2 be rays in F ∗ . Since both R1 , R2 ∈ ∈ (G), R1 , respectively R2 , must have an infinite number of vertices in some Xi , respectively Xj . Hence by Lemma 16, there is a infinite collection of disjoint paths linking R1 and R2 in F ∗ , and so R1 and R2 cannot be finitely separated in F ∗ . Hence, R1 and R2 are contained in the same end of F ∗ ; i.e., 1 = 2 . 

4.

PROOF OF THEOREM 1

Proof. Construct an end-faithful, spanning eulerian subgraph F ∗ of G as in Section 3. By Theorem 6, from such a subgraph one obtains a topological Euler tour  : S1 → |F ∗ | ⊆ |G| which is injective on (F ∗ ) = (G). Since  is a closed dominating trail of G injective on (G), by Lemma 10, L(G) is hamiltonian.  The proof above clearly depends on having a finite number of ends, and requires each of them to be thin. It is relatively straightforward to construct an example of a locally finite graph with a single thick end, where the method above fails. Neither is it difficult to construct a locally finite graph with infinitely many thin ends, in which the above method similarly fails. One might hope to refine the technique for these cases, either in the initial selection of F or in the addition of crossing edges. We have recently received news that F. Lehner has extended Theorem 1 to include locally finite graphs with finitely many ends, some of which may be thick [26]. The problem remains open for graphs with infinitely many ends. This remains an interesting problem for further study. It would also be interesting to know if the requirement of 6-edge-connected can be relaxed to 4-edge-connected, as in the finite case. Journal of Graph Theory DOI 10.1002/jgt

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REFERENCES [1] C. Paul Bonnington and R. Bruce Richter, Graphs embedded in the plane with a bounded number of accumulation points, J Graph Theory 44(2) (2003), 132–147. [2] H. Bruhn, The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits, J Combin Theory Ser B 92(2) (2004), 235–256. [3] H. Bruhn, R. Diestel, and M. Stein, Cycle-cocycle partitions and faithful cycle covers for locally finite graphs, J Graph Theory 50(2) (2005), 150–161. [4] H. Bruhn, R. Diestel, and M. Stein, Menger’s theorem for infinite graphs with ends, J Graph Theory 50(3) (2005), 199–211. [5] H. Bruhn and M. Stein, MacLane’s planarity criterion for locally finite graphs, J Combin Theory Ser B 96(2) (2006), 225–239. [6] H. Bruhn and M. Stein, On end degrees and infinite cycles in locally finite graphs, Combinatorica 27(3) (2007), 269–291. [7] H. Bruhn and X. Yu, Hamilton cycles in planar locally finite graphs, SIAM J Discrete Math 22(4) (2008), 1381–1392. [8] K. Casteels and R. Bruce Richter, The bond and cycle spaces of an infinite graph, J Graph Theory 59(2) (2008), 162–176. [9] Q. Cui, J. Wang, and X. Yu, Hamilton circles in infinite planar graphs, J Comb Theory, Ser B 99(1) (2009), 110–138. [10] R. Diestel, The cycle space of an infinite graph, Comb Probab Comput 14(1–2) (2005), 59–79. [11] R. Diestel, Graph theory, Graduate Texts in Mathematics, vol. 173, Springer, Berlin, 2005. [12] R. Diestel, Locally finite graphs with ends: a topological approach. http://www.citebase.org/abstract?id=oai:arXiv.org:0912.4213, 2009. [13] R. Diestel and D. K¨uhn, On infinite cycles. I, Combinatorica 24(1) (2004), 69–89. [14] R. Diestel and D. K¨uhn, On infinite cycles. II, Combinatorica 24(1) (2004), 91–116. [15] R. Diestel and D. K¨uhn, Topological paths, cycles and spanning trees in infinite graphs, Eur J Comb 25(6) (2004), 835–862. [16] H. Fleischner, The square of every two-connected graph is Hamiltonian, J Comb Theory Ser B 16 (1974), 29–34. [17] A. Georgakopoulos, Fleischner’s theorem for infinite graphs, Oberwolfach Reports 4 (2007). [18] A. Georgakopoulos, Infinite hamilton cycles in squares of locally finite graphs, Adv Math 220(3) (2009), 670–705. [19] A. Georgakopoulos, Topological circles and Euler tours in locally finite graphs, Electron J Comb 16(1) (2009), Research Paper 40, 16. [20] A. Georgakopoulos and P. Spr¨ussel, Geodetic topological cycles in locally finite graphs, Electron J Comb 16(R144) (2009), 1. Journal of Graph Theory DOI 10.1002/jgt

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[21] F. Harary and C. St. J. A. Nash-Williams, On Eulerian and Hamiltonian graphs and line graphs, Can Math Bull 8 (1965), 701–709. [22] C. Thomassen, Hamiltonian paths in squares of infinite locally finite blocks, Ann Discrete Math 3 (1978), 269–277. Advances in Graph Theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). [23] C. Thomassen, Reflections on graph theory, J Graph Theory 10(3) (1986), 309–324. [24] W. T. Tutte, A theorem on planar graphs, Trans Am Math Soc 82(1) (1956), 99–116. [25] A. Vella and R. Bruce Richter, Cycle spaces in topological spaces, J Graph Theory 59(2) (2008), 115–144. [26] F. Lehner, The line graph of every locally finite 6-edge-connected graph with finitely many ends is hamiltonian, M.Sc. thesis, Graz University of Technology, 2011.

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results can be used to choose the chaotic generator more suitable for applications on chaotic digital communica- .... faster is its split from other neighboring orbits [10]. This intuitively suggests that the larger the Lyapunov number, the easier sh

Invariant random subgroups of locally finite groups
Observation. Suppose that G r (Z,µ) is a measure-preserving action on a probability space. Let f : Z → SubG be the G-equivariant map defined by z ↦→ Gz = {g ...

pdf-1294\structure-of-decidable-locally-finite-varieties-progress-in ...
... the apps below to open or edit this item. pdf-1294\structure-of-decidable-locally-finite-varietie ... -in-mathematics-by-ralph-mckenzie-matthew-valeriote.pdf.

Exploiting the Unicast Functionality of the On ... - Semantic Scholar
any wired infrastructure support. In mobile ad hoc networks, unicast and multicast routing ... wireless network with no fixed wired infrastructure. Each host is a router and moves in an arbitrary manner. ..... channel reservation control frames for u

the impact of young workers on the aggregate ... - Semantic Scholar
years, from various years of the Statistical Abstract of the United States. The following observations are missing: 16–19 year olds in 1995 and 1996; 45–54 year old men in Utah in 1994; and women 65 and over in. Delaware, Idaho, Mississippi, Tenn

the impact of young workers on the aggregate ... - Semantic Scholar
An increase in the share of youth in the working age population of one state or region relative to the rest of the United States causes a sharp reduction in that state's relative unemployment rate and a modest increase in its labor force participatio

Exploiting the Unicast Functionality of the On ... - Semantic Scholar
Ad hoc networks are deployed in applications such as disaster recovery and dis- tributed collaborative computing, where routes are mostly mul- tihop and network ... the source must wait until a route is established before trans- mitting data. To elim

The impact of host metapopulation structure on the ... - Semantic Scholar
Feb 23, 2016 - f Department of Biology and Biochemistry, University of Bath, Claverton Down, Bath, UK g Center for ... consequences of migration in terms of shared genetic variation and show by simulation that the pre- viously used summary .... is se

On Knowledge - Semantic Scholar
Rhizomatic Education: Community as Curriculum by Dave Cormier. The truths .... Couros's graduate-level course in educational technology offered at the University of Regina provides an .... Techknowledge: Literate practice and digital worlds.

On Knowledge - Semantic Scholar
Rhizomatic Education: Community as Curriculum .... articles (Nichol 2007). ... Couros's graduate-level course in educational technology offered at the University ...

The effect of disfluency on mind wandering during ... - Semantic Scholar
United States b University of Notre Dame, Department of Computer Science .... findings and those for tasks like list learning suggests that disfluency only affects surface level learning (e.g. ... The experiment utilized a between-subjects design.

On the Use of Variables in Mathematical Discourse - Semantic Scholar
This is because symbols have a domain and scope ... In predicate logic, the symbol x is considered a free ... x has to be considered a free variable, given the ab-.