5.5 Subgraphs induced by all the contractible
there is a partition (X, X0) of V(G\x)such that E(X, X0) is non-empty and all edges in E(X, X0) are non-contractible. Then G contains infinitely many disjointX-X0 edges: x0x00, x1x01, . . . such that xix0i converges to an end ω of G with ω6=x.
Proof. Letx0x00be aX-X0edge andS0 be a 3-separator containingx0 andx00. LetC0 be a component ofG−S0 such thatx /∈C0 andC00 be a component of G−S0such that x∈C00 ∪S0. By Lemma 5.21, there exists contractible edges x0y0 andx00y00 withy0∈X∩C0 andy00∈X0∩C0. Note that y06=y00. Choose a pathQ0inC0 betweeny0andy00. Then there exists aX-X0 edgex1x01inQ0. LetS1 be a 3-separator containingx1and x01. LetC10 be the component of G−S1so thatC10∩ {x0, x00} 6=∅andC1be any component ofG−S1other than C10. Then{x0, x00} ⊆C10 ∪S1. By Lemma 5.21, there exists contractible edges x1y1 and x01y01 with y1 ∈ X∩C1 and y10 ∈ X0∩C1. Choose a path Q1 in C1
betweeny1 andy01. Then there exists aX-X0 edgex2x02 inQ1. Note thatx2x02 is disjoint fromx0x00andx1x01. Letz10∈C10∩ {x0, x00}. Then any two internally disjoint paths betweenz01andx2 must meet{x1, x01}.
LetS2 be a 3-separator containingx2and x02. LetC20 be the component of G−S2 so that C20 ∩ {x1, x01} 6=∅ and C2 be any component of G−S2 other thanC20. Then{x1, x01} ⊆C20 ∪S2. SinceGis 3-connected,G[C2∪ {x2, x02}] is 2-connected. Because any two internally disjoint paths betweenz10 andx2must meet{x1, x01}, z01 ∈/ C2. If z10 ∈C20, then {x0, x00} ⊆C20 ∪S2. If z10 ∈S2, then there can only be one z01−C2 edge for otherwise there will be two internally disjoint paths betweenz01andx2 not intersecting{x1, x01}. Thisz10−C2edge is contractible by Lemma 5.21. Sincex0x00is non-contractible,{x0, x00} ⊆C20∪S2. By Lemma 5.21, there exists contractible edgesx2y2andx02y20 withy2∈X∩C2 and y02 ∈ X0 ∩C2. Choose a path Q2 in C2 betweeny2 and y20. Then there exists aX-X0 edgex3x03 inQ2. Note thatx3x03 is disjoint fromx0x00,x1x01 and x2x02. Letz02∈C20 ∩ {x0, x00}andz12∈C20∩ {x1, x01}. Then fori= 0,1, any two internally disjoint paths betweenzi2 andx3 must meet{x2, x02}.
Suppose we have constructed pairwisely disjointX-X0edgesx0x00, x1x01, . . . , xnx0n together with:
1. a 3-separatorSn−1 containingxn−1andx0n−1,
2. a component Cn−10 of G−Sn−1 such that x0, x00, x1, x01, . . . , xn−2, x0n−2 ∈ Cn−10 ∪Sn−1,
3. a componentCn−1ofG−Sn−1 other thanCn−10 such thatxn, x0n∈Cn−1, 4. vertices yn−1 andyn−10 inCn−1 such thatxn−1yn−1 andx0n−1y0n−1 are
con-tractible edges,
5. a pathQn−1inCn−1betweenyn−1andy0n−1containing anX-X0edgexnx0n, and
6. verticeszn−10 , z1n−1, . . . , zn−1n−2such thatzn−1i ∈Cn−10 ∩{xi, x0i}fori= 0,1, . . . , n−
2.
Let Sn be a 3-separator containing xn and x0n. Let Cn0 be the component of G−Sn so that Cn0 ∩ {xn−1, x0n−1} 6= ∅ and Cn be any component of G− Sn other than Cn0. Then {xn−1, x0n−1} ⊆ Cn0 ∪Sn. Since G is 3-connected, G[Cn ∪ {xn, x0n}] is 2-connected. For each i = 0,1, . . . , n−2, because any two internally disjoint paths between zin−1 and xn must meet {xn−1, x0n−1}, zin−1 ∈/ Cn. If zin−1 ∈ Cn0, then {xi, x0i} ⊆ Cn0 ∪Sn. If zin−1 ∈ Sn, then {xn−1, x0n−1} ⊆ Cn0. If there are at least two zin−1−Cn edges, then since G[Cn∪{xn, x0n}] is 2-connected, we can find two internally disjoint paths between zin−1 andxn not meeting{xn−1, x0n−1}, a contradiction. Thus, there can only be onezin−1−Cnedge which is contractible by Lemma 5.21. Sincexix0i is non-contractible,{xi, x0i} ⊆Cn0 ∪Sn. By Lemma 5.21, there exists contractible edges xnyn andx0nyn0 withyn ∈X∩Cn and y0n∈X0∩Cn. Choose a pathQn in Cn
betweenynandyn0. Then there exists aX-X0 edgexn+1x0n+1 inQn. Note that xn+1x0n+1 is disjoint from x0x00, x1x01, . . . , xnx0n. For each i = 0,1, . . . , n−1, since{xi, x0i} ⊆Cn0 ∪Sn, we can choosezni ∈Cn0 ∩ {xi, x0i}.
By applying Lemma 7.17 to U :={x0, x1, . . . ,}, there exists a ray R such that there are infinitely many disjointU-Rpaths. Let ω be the end containing R. Without loss of generality, we may assumex0x00, x1x01, . . .converges toω. If xis not an end, then clearly, ω 6=x. Suppose xis an end. Then x∈C00. By construction,x0, x00, x1, x01 ∈C0∪S0. ConsiderQ01:=x1y1∪Q1∪x01y10. Since {x0, x00} ⊆C10 ∪S1,Q01∩ {x0, x00}=∅. Therefore,{x2, x02} ⊆C0∪S0. Suppose we have shown that {xn, x0n} ⊆ C0∪S0. Consider Q0n :=xnyn ∪Qn∪x0nyn0. Since{x0, x00} ⊆Cn0∪Sn,Q0n∩{x0, x00}=∅. Therefore,{xn+1, x0n+1} ⊆C0∪S0. By induction,{xi, x0i} ⊆C0∪S0for alli= 0,1,2, . . .. Hence,ω∈C0andω6=x.
We are now ready for the proof of Theorem 5.3.
Theorem 5.3. Let G be a 3-connected locally finite infinite graph which is triangle-free or has minimum degree at least4, andGC be the subgraph induced by all the contractible edges. ThenGC is topologically 2-connected.
Proof. SupposeGC is not topologically 2-connected. Then there exists a point x in GC such that GC \x is not connected. Let U and U0 be two disjoint non-empty open sets in |G| such that GC \x ⊆ U ∪U0, (GC \x)∩U 6= ∅ and (GC\x)∩U0 6= ∅. Define X := (GC\x)∩U ∩V(G) and X0 := (GC\ x)∩U0 ∩V(G). Since GC is a spanning subgraph of G by Corollary 5.22, X ∪X0 = V(G\x). By the connectedness of an edge and the definition of a basic open neighborhood of an end, both X and X0 are non-empty. Since G\xis topologically connected,E(X, X0) is non-empty. Supposexis a vertex or an end of G. By the connectedness of an edge, all edges in E(X, X0) are non-contractible. Supposexis an interior point of an edgee. Then all edges in E(X, X0) are non-contractible unlesse∈E(X, X0)∩EC. Note that in this case, E(X, X0)−eis non-empty asGis 2-connected, and every edge inE(X, X0)−eis non-contractible. Lete=yy0 where y∈X andy0 ∈X0. Suppose X ={y}. By
Corollary 5.22,y is incident to at least two contractible edges, a contradiction.
Therefore, we can use y instead ofxand (X−y, X0) as the required partition ofV(G\y) in Lemma 5.24.
By Lemma 5.24,Gcontains infinitely many disjointX-X0edges: x0x00, x1x01, . . . such thatxix0i converges to an endω ofGwith ω6=x. Since GC is spanning, GC\x contains all the ends of |G| except possibly x. Without loss of gener-ality, assume ω ∈U. SinceU is open, there exists a basic open neighborhood C(S, ω)ˆ ⊆U. Sincex00, x01, x02. . . ,∈ U0 converge to ω, infinitely many of them
lie in ˆC(S, ω) contradictingU∩U0=∅.
Note that Theorem 5.3 is best possible as demonstrated by the cartesian product of a ray and a triangle which contains a triangle and has minimum degree 3. Here, GC contains 3 disjoint rays together with their common end, and is connected but not topologically 2-connected.
Lemma 5.25 (Diestel and K¨uhn [12]). Let G is a locally finite graph. Then every closed connected subspace of|G| is arc-connected.
Corollary 5.26. Let G be a 3-connected locally finite infinite graph which is triangle-free or has minimum degree at least 4. Then every contractible edge of Glies in a circle consisting of contractible edges.
Proof. Letxybe a contractible edge inG. SinceGCis topologically 2-connected, GC−xyis connected. By Lemma 5.25,GC−xyis arc-connected and there is an arc A between xand y in GC−xy. Then A∪xy is a circle consisting of
contractible edges.
Unfortunately, the arguments used in the proof of Theorem 5.3 cannot be generalized directly to k-connected locally finite infinite graphs fork ≥4. We end this paper with the following conjecture.
Conjecture 5.27. Let G be a k-connected locally finite infinite graph (k≥4) which is triangle-free or has minimum degree greater than 32(k−1), and GC be the subgraph induced by all the contractible edges. Then GC is topologically 2-connected.
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Chapter 6
Contraction-critical
4-connected locally finite infinite graphs
6.1 Introduction
It is well-known that the only contraction-critical 2-connected and 3-connected finite graphs areK3andK4respectively. For 4-connected graphs, Fontet [6] and Martinov [8] independently proved that every contraction-critical 4-connected finite graph is either the square of a cycle or the line graph of a cyclically 4-edge-connected cubic graph. This is equivalent to the characterization that a 4-connected finite graph is contraction-critical if and only if it is 4-regular and every edge lies in a triangle (see also Martinov [9]).
Recently, Ando and Egawa [1] proved that for any vertex of degree greater than four in a 4-connected finite graph, there exists a contractible edge at dis-tance one or less from that vertex. They [2] also showed that for every non-contractible edge not lying in a triangle, there exists a non-contractible edge at distance one or less from that edge.
For infinite graphs, Kriesell [7] gave a construction of contraction-critical k-connected graphs fork≥2 such that all vertices have infinite degree. On the other hand, the author [6, 7] showed that fork= 2,3, everyk-connected locally finite infinite graph contains infinitely many contractible edges. In this paper, we modify the results in Ando and Egawa’s paper [1] slightly and extend Fontet and Martinov’s result to locally finite infinite graphs.
Theorem 6.1. A 4-connected locally finite graph is contraction-critical if and only if it is4-regular and every edge lies in a triangle.
Theorem 6.2. A4-connected locally finite infinite graph is contraction-critical if and only if it is the line graph of a 3-edge-connected and cyclically 4-edge-connected cubic graph.
Moreover, a theorem of Ando and Egawa [1] on the lower bound of con-tractible edges can be generalized to 4-connected locally finite graphs.
Theorem 6.3. Every 4-connected locally finite graphG has at least |V≥5(G)|
contractible edges.