Group Connectivity of 3-Edge-Connected Chordal Graphs

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1 Gaphs and Combinatoics (2000) 16 : 165±176 Gaphs and Combinatoics ( Spinge-Velag 2000 Goup Connectivity of 3-Edge-Connected Chodal Gaphs Hong-Jian Lai Depatment of Mathematics, West Viginia Univesity, Mogantown, WV 26506, USA Abstact. Let A be a nite abelian goup and G be a digaph. The bounday of a function f : E G 7!A is a function q f : V G 7!A given by q f v ˆPe leaving v f e P Pe enteing v f e. The gaph G is A-connected if fo evey b : V G 7!A with v A V G b v ˆ0, thee is a function f : E G 7!A f0g such that q f ˆ b. In [J. Combinatoial Theoy, Se. B 56 (1992) 165±182], Jaege et al showed that evey 3-edgeconnected gaph is A-connected, fo evey abelian goup A with jaj V 6. It is conjectued that evey 3-edge-connected gaph is A-connected, fo evey abelian goup A with jaj V 5; and that evey 5-edge-connected gaph is A-connected, fo evey abelian goup A with jaj V 3. In this note, we investigate the goup connectivity of 3-edge-connected chodal gaphs and chaacteize 3-edge-connected chodal gaphs that ae A-connected fo evey nite abelian goup A with jaj V Intoduction We conside nite gaphs which may contain loops o multiple edges. The goups consideed in this pape ae nite Abelian (additive) goups. Fo a nite Abelian goup A, the additive identity of A will be denoted by 0 (zeo) thoughout this pape. Let G and H be gaphs. If H is a subgaph of G, then we wite H J G. Let G be a digaph. Fo a vetex v A V G, let EG v ˆf u; v A E G : u A V G g; and E G v ˆf v; u A E G : u A V G g: The subscipt G may be omitted when G is undestood fom the context. Let A be a nontivial Abelian goup and let A denote the set of nonzeo elements in A. De ne F G; A ˆff : E G!Ag and F G; A ˆff : E G!A g: Fo each f A F G; A, the bounday of f is a function q f : V G!A de ned by q f v ˆ f e f e ; e A E v e A E v whee ``P'' efes to the addition in A. Thoughout this pape, we shall adopt the

2 166 H.-J. Lai following convenience: if J E G and if f :! A is a function, then we egad f as a function f : E G!A such that f e ˆ0 fo all e A E G. Let G be an undiected gaph and A be an abelian goup. A function b : V G!A is called an A-valued zeo sum function on G if P v A V G b v ˆ0in G. The set of all A-valued zeo sum functions on G is denoted by Z G; A. A gaph G is A-connected if G has an oientation G 0 such that fo evey function b A Z G; A, thee is a function f A F G 0 ; A such that b ˆ q f. Fo an Abelian goup A, let hai denote the family of gaphs that ae A-connected. It is obseved in [7] that that G AhAi is independent of the oientation of G. An A-nowhee-zeo- ow (abbeviated as A-NZF) in G is a function f A F G; A such that q f ˆ 0. The nowhee-zeo- ow poblems wee intoduced by Tutte [14], and ecently suveyed by Jaege in [6]. The concept of A-connectivity was intoduced by Jaege et al in [7], whee A- NZF's wee successfully genealized to A-connectivities. A concept simila to the goup connectivity was independently intoduced in [8], with a di eent motivation fom [7]. Tutte's 3- ow conjectue ([2], Unsolved Poblem 48) states that evey 4-edgeconnected gaph has a Z 3 -NZF. It is known that plana gaphs and pojective plana gaphs ae two gaph families in which Tutte's 3- ow conjectue holds ([12] and [16]). In this note, we shall show (in Section 4) that evey 4-edge-connected chodal gaph is A-connected, fo each Abelian goup A with jaj V 3. In paticula, evey 4-edge-connected chodal gaph admits a Z 3 -NZF. Thee ae examples, pesented in Section 4, showing that both being chodal and being 4-edge-connected cannot be elaxed in this esult. We also investigate 3-edge-connected chodal gaphs in Section 4 and chaacteize those 3-edge-connected chodal gaphs which ae not Z 3 -connected. In Section 2, fome elated esults ae pesented and in Section 3, some popeties on goup connectivity will be developed. These popeties will be used in Section 4 to pove the main esults. 2. Pio Results We pesent some pio esults on Z 3 -NZF's in this section. Theoem 2.1 (GoÈtzsch [4]). Evey 4-edge-connected plana gaph has a Z 3 -NZF. Theoem 2.2 (Tutte [15]). Let G be a 3-egula gaph. Then G has a Z 3 -NZF if and only if G is bipatite. Theoem 2.3 (GuÈbaum [5], Aksionov [1], Steinbeg and Younge [11] and Thomassen [13]). Evey 2-edge-connected gaph with at most thee edge cuts of size 3 and embedable in the plane has a Z 3 -NZF. Theoem 2.4 (Steinbeg and Younge [11]). Evey 2-edge-connected gaph with at most one edge cut of size 3 and embedable in the pojective plane has a Z 3 -NZF.

3 Goup Connectivity of 3-Edge-Connected Chodal Gaphs 167 Theoem 2.5 (Lai and Zhang [10]). Let G be a k-edge-connected gaph with t odd vetices. If k V 4dlog 2 te, then G has a Z 3 -NZF. It is conjectued in [7] that evey 5-edge-connected gaphs is in hz 3 i. 3. Some Popeties We need the notion of gaph contaction. Let G be a gaph and let J E G be an edge subset. The contaction G= is the gaph obtained fom G by identifying the two ends of each edge e in and deleting e. If ˆfeg, then we wite G=e fo G=feg. IfH is a subgaph of G, then we wite G=H fo G=E H. Note that even G is a simple gaph, the contaction G= may have loops and multiple edges. Let G be a gaph and let v A V G be a vetex of degee m V 4. Let N v ˆfv 1 ; v 2 ;...; v m g denote the set of vetices adjacent to the vetex v, and let ˆfvv 1 ; vv 2 g. The gaph G v; Š is obtained fom G fvv 1 ; vv 2 g by adding a new edge that joins v 1 and v 2.Ifm ˆ 2k is even and if M ˆ hfv 1 ; v k 1 g; fv 2 ; v k 2 g;...; fv k ; v 2k gi is a way to pai the vetices in N v, theng v; M denotes the gaph obtained fom G v by adding k new edges e i joining v i and v k i ; 1 U i U k. Lemma 3.1. Let A be an Abelian goup. Let G be a gaph and let v A V G be a vetex of degee m V 4. (i) If fo some of two edges incident with v in G, G v; Š AhAi, then G AhAi. (ii) Let m be even, let M be a way to pai the vetices of N v such that G v; M AhAi, and let b A Z G; A be given. If b v ˆ0 A A, then thee is a function f A F G; A such that q f ˆ b. (iii) (Coollay 2.3, [7]) Let e ˆ v 1 v 2 be an edge in G. If G e AhAi, then G AhAi. Poof. (i). We may assume that N v ˆfv 1 ;...; v m g and ˆfvv 1 ; vv 2 g. Since being A-connected is a popety independent of the oientation, we may assume that in G, the two edges in ae oiented fom v 1 to v, and fom v to v 2 ; and we may assume that in G v; Š, the newly added edge (denoted by v 1 v 2 ) is oiented fom v 1 to v 2. Let b A Z G; A. Since V G ˆV G v; Š, b A Z G v; Š ; A also. Since G v; Š AhAi, thee is a function f 0 A F G v; Š ; A such that q f 0 ˆ b. De ne f A F G; A by f e ˆ f 0 e if e A E G, and f v 1 v ˆ f vv 2 ˆ f 0 v 1 v 2. Then it is easy to see that f A F G; A such that q f ˆ b. This poves (i). (ii). The poof fo (ii) is simila to that fo (i), and so is omitted. Fo a subgaph H of a gaph G, let A G H denote the vetices in V H that ae adjacent to some vetices in V G V H in G. (Vetices in A G H ae called the vetices of attachment of H in G.) Poposition 3.2. Fo any Abelian goup A, hai is a family of connected gaphs

4 168 H.-J. Lai satisfying each of the following holds: (C1) K 1 AhAi, (C2) if e A E G and if G AhAi, then G=e AhAi, and (C3) if H AhAi and if G=H AhAi, then G AhAi. (A family of connected gaphs satisfying (C1)±(C3) is called a complete family, st intoduced by Catlin [3]. In fact, Catlin de ned and studied complete families in a moe geneal way. Fo moe popeties of complete families, see [3].) Poof. Let A be an Abelian goup. By Poposition 2.1 of [7], evey membe in hai is connected. Note that (C1) follows fom the de nition of hai tivially. Let e A E G and assume that G AhAi. Let G 0 be G=e with an oientation, let the two ends of e be u and v and let w denote the vetex in G 0 to which u and v ae identi ed. Let G also denote the digaph with the same oientation as G 0 on the edges in E G feg and with e oiented fom u to v. Fo any b 0 A Z G 0 ; A, de ne a function b by 8 b 0 z if z A V G 0 fwg ˆV G fu; vg >< b z ˆ b 0 w if z ˆ u >: 0 if z ˆ v. Then P z A V G b z ˆPz A V G 0 b0 z ˆ0, and so b A Z G; A. Since G AhAi, thee is a function f A F G; A with q f ˆ b. Let f 0 be the estiction of f to E G feg. Since q f 0 w ˆ f e 0 f e 0 e 0 A E G v U E G u feg e 0 A E G v U E G u feg ˆ q f u q f v ˆb u b v ˆb 0 w ; q f 0 ˆ b 0 and so by de nition, G=e AhAi. Theefoe (C2) holds. Suppose that both H AhAi and G=H AhAi. We may assume that G has a xed oientation. Thus the edges in both H and G=H ae oiented by the oientation of G. By Lemma 3.1(iii), we may assume that H is an induced subgaph of G, and so E G is the disjoint union of E H and E G=H. Note that H is connected and so H will be contacted to a vetex v H (say) in G=H. Let b A Z G; A and let a 0 ˆ Pv A V H b v. De ne b 1 : V G=H!A by ( b 1 z ˆ b z if z 0 v H a 0 if z ˆ v H. Then P z A V G=H b 1 z ˆPz A V G b z and so b 1 A Z G=H; A. Since G=H AhAi, thee is a function f 1 A F G=H; A such that q f 1 ˆ b 1. Fo each z A V H, de ne ( b z P e A E G=H b 2 z ˆ v H V E G z f 1 e P e A E G=H v H V E G z f 1 e if z A A G H b z othewise.

5 Goup Connectivity of 3-Edge-Connected Chodal Gaphs 169 Note that b 2 z ˆ z A V H z A V H b z e A E G=H v H f 1 e e A E G=H v H f 1 e ˆa 0 q f 1 v H ˆ0; and so b 2 A Z H; A. Since H AhAi, thee is a function f 2 A F H; A such that q f 2 ˆ b 2. De ne, fo each e A E G, f e ˆ f 1 e f 2 e. Since E G is the disjoint union of E H and E G=H, f A F G; A ; and fo a vetex z A V G, q f z ˆq f 1 z q f 2 z ˆq f 1 z b z q f 1 z ˆb z : Theefoe G AhAi, and so (C3) follows. The ``only if'' pat of Lemma 3.3 was obseved in [7] without a poof. We pesent the whole poof of Lemma 3.3 fo the sake of completeness. Lemma 3.3. Let n V 1 denote an intege and let C n denote the cycle of n vetices. Then fo any Abelian goup A, C n AhAi if and only if jaj V n 1. Poof. Let C n ˆ v 1 v 2 v n v 1 and assume that the edge v i v i 1 is oiented fom v i to v i 1, fo each i ˆ 1; 2;...; n mod n. Suppose st that C n AhAi. We shall assume jaj ˆm U n to nd a contadiction. Let A ˆfa1 0 ; a0 2 ;...; a0 m g with a0 m ˆ 0. Let a 0 ˆ 0, a 1 ˆ a1 0 and a i ˆ ai 0 a0 i 1 fo 2 U i U m 1 and a m ˆ P m 1 iˆ1 a i. Then a 1 ; a 2 ;...; a m 1 is a sequence of elements in A such that f P i jˆ1 a j : 1 U i U m 1g ˆA f0g. Moeove, fo any x A A, ( ) x i a j : 1 U i U m 1 ˆ A: 1 jˆ0 P De ne b v i ˆa i fo 1 U i U m, and b v i ˆ0 fo m 1 U i U n. Since jaj ˆm, n iˆ1 b v i ˆPm 1 iˆ1 a i a m ˆ 0 A A and so b A Z C n ; A. Since C n AhAi, thee is a function f A F C n ; A such that q f ˆ b. Denote f v n v 1 by x. Then since q f ˆ b, we must have f v i v i 1 ˆx P i jˆ1 a i fo all i ˆ 1; 2;...; m and f v i v i 1 ˆx fo all i ˆ m 1;...; n. By (1), one of f v i v i 1, whee 1 U i U n, must be 0 A A, contay to the assumption that f A F C n ; A. Convesely, assume that jaj V n 1. Let b A Z C n ; A, and let B ˆfaA A : a ˆ b v i fo some 1 U i U n 1g. Then jbj U n 1. Since jaj V n 1, thee is an x A A BUf0g. De ne f A F G; A by f v i v i 1 ˆb v i x (1 U i U n 1 and f v n v 1 ˆx. Then q f v i ˆb v i and f A F C n ; A. Hence C n AhAi. Let k V 1 be an intege and let H be a subgaph of G. The k-closue of H in G is the (unique) maximal subgaph of G the fom H U G 1 U U G n whee fo each i, 1U i U n, G i is a cycle and je G i E H U G 1 U U G i 1 j U k. Coollay 3.4 below follows fom Lemma 3.3 and Poposition 3.2(C3).

6 170 H.-J. Lai Coollay 3.4 (Coollay 2.4 of [7]). Let A be a nite Abelian goup with jaj > k. Let H be a subgaph of G. If H AhAi, then the k-closue of H in G is also in hai. Coollay 3.5 below follows fom Poposition 3.2(C3) and Lemma 3.3. Coollay 3.5. Let A be an Abelian goup with jaj V 3, let n V 5 be an intege, and let K n e denote the gaph obtained fom K n by emoving an edge fom K n. Each of the following holds: (i) K n e AhAi, and (ii) K n AhAi. Poposition 3.6. Let W n be the wheel of n 1 vetices. Then W 4 AhAi, fo any Abelian goup A with jaj V 3. Poof. Let v 1 v 2 v 3 v 4 v 1 be the im 4-cycle of W 4 and let v denote the cente vetex of W 4.IfjAjV4, then by Lemma 3.3 and Poposition 3.2, W 4 AhAi. Hence we only need to pove the case when A ˆ Z 3. Expess Z 3 ˆf0; 1; 1g. Let b A Z W 4 ; Z 3. We shall nd f A F W 4 ; Z 3 by de ning f e ˆ1; Ee A E W 4 and adjust the oientation of W 4 to meet the equiement of q f ˆ b. In the est of the poof, fo an edge e ˆ xy A E W 4, we wite x; y to mean that e is oiented fom x to y. An oientation D of W 4 will be expessed by a set of oiented edges. If b v ˆ0, then let M ˆ hfv 1 ; v 2 g; fv 3 ; v 4 gi be a patition of N v. Then W 4 v; M is the 3-egula gaph with a 4-cycle and two disjoint 2-cycles. By Lemma 3.3 and by Poposition 3.2(C3), W 4 v; M AhZ 3 i, and so by Lemma 3.1(ii), thee is a function f A F W 4 ; Z 3 such that q f ˆ b. Hence we assume that b v 0 0 A Z 3, and so b v A f1; 1g. We only need to show that case when b v ˆ1, by symmety. Since P z A V W 4 b z ˆ0, we may assume that b v A Z 3. Suppose st that b v 1 ˆ 1. Since b A Z W 4 ; Z 3, eithe b v 2 ˆb v 3 ˆ b v 4 ˆ1, whence D 1 ˆf v; v 1 ; v 2 ; v ; v 3 ; v ; v 4 ; v ; v 1 ; v 2 ; v 2 ; v 3 ; v 3 ; v 4 ; v 4 ; v 1 g is an oientation satisfying q f ˆ b; ob v 2 ˆb v 3 ˆb v 4 ˆ0, whence D 2 ˆf v 1 ; v ; v 2 ; v ; v; v 3 ; v 4 ; v ; v 2 ; v 1 ; v 2 ; v 3 ; v 4 ; v 3 ; v 4 ; v 1 g is an oientation satisfying q f ˆ b. Hence we assume that b v 1 ˆ1, and by symmety, we assume that b v i 0 1; Ei with 2 U i U 4. Since b A Z W 4 ; Z 3 and by symmety, eithe b v 2 ˆ1 and b v 3 ˆb v 4 ˆ0, whence D 3 ˆf v 1 ; v ; v 2 ; v ; v; v 3 ; v 4 ; v ; v 1 ; v 2 ; v 2 ; v 3 ; v 4 ; v 3 ; v 4 ; v 1 g is an oientation satisfying q f ˆ b; ob v 3 ˆ1 and b v 2 ˆb v 4 ˆ0, whence D 4 ˆf v 1 ; v ; v 2 ; v ; v 3 ; v ; v; v 4 ; v 2 ; v 1 ; v 2 ; v 3 ; v 3 ; v 4 ; v 1 ; v 4 g is an oientation satisfying q f ˆ b.

7 Goup Connectivity of 3-Edge-Connected Chodal Gaphs 171 In any case, such a function f A F W 4 ; Z 3 can be found so that q f ˆ b, and so W 4 AhZ 3 i, by de nition. 4. Goup Connectivity of Chodal Gaphs We shall pove the following Theoem 4.2, which implies that evey 4-edgeconnected chodal gaph has a Z 3 -NZF. A gaph G is chodal if evey induced cycle of G has length at most 3. Lemma 4.1 is an easy obsevation. Lemma 4.1. Evey 2-connected simple gaph H with jv H j V 4 has a cycle of length at least 4. Theoem 4.2. Let A be an Abelian goup of ode at least 3. If G is a 4-edgeconnected chodal gaph, then G AhAi. Befoe poving Theoem 4.2, we pesent some examples to show that both conditions of G in Theoem 4.2 ae needed. Example 4.3. We shall show that being 4-edge-connected cannot be elaxed in Theoem 4.2, in the sense that thee ae in nitely many 3-edge-connected chodal gaphs that ae not in hz 3 i. The gaph K 4 is a 3-edge-connected chodal gaph that does not have a Z 3 -NZF, and so it cannot be Z 3 -connected. By Poposition 3.2(C2), any gaph contactible to K 4 cannot be Z 3 -connected. In paticula, any gaph with a block (a maximal 2-connected subgaph) isomophic to K 4 cannot be Z 3 -connected. In fact, thee is a class of 3-edge-connected, 2-connected chodal gaphs that ae not Z 3 -connected. We shall pesent this class. Let m V 1 be an intege and let G 1 ; G 2 ;...; G m be m disjoint copies of K 4 each of which has a distinguished edge e i ˆ x i y i, 1 U i U m. Let G m denote the gaph obtained fom the disjoint union of G 1 e 1 ; G 2 e 2 ;...; G m 1 e m 1 and G m by identifying x 1 ; x 2 ;...; x m into a single vetex x and by identifying y 1 ; y 2 ;...; y m into a single vetex y. The edge e m, with its ends being x and y, is now an edge in G m. We shall show that G m does not have a Z 3 -NZF, and theefoe cannot be Z 3 -connected. Suppose that G m has a Z 3 -NZF. Thus G m is oiented and thee is a function f A F G m ; Z 3 such that q f ˆ 0. Expessing Z 3 ˆf0; 1; 1g and evesing the diection of the edges in G m if necessay, we may assume that f e ˆ1, Ee A E G m. Fo each i with 1 U i U m, denote V G i ˆfx i ; y i ; u i ; v i g, whee u i and v i ae the two vetices of this subgaph G i with degee 3 in G m. Since f e ˆ1; Ee A E G m, we may assume that all thee edges incident with u i ae diected out fom u i, and so all thee edges incident with v i ae diected into v i, fo each i with 1 U i U m. It follows by q f ˆ 0that f e m ˆ0, contay to the assumption that f A F G m ; Z 3. Example 4.4. Being chodal cannot be elaxed in Theoem 4.2 eithe, in the sense that thee ae 4-edge-connected non-chodal gaphs that ae not Z 3 -connected. The following class of gaphs (togethe with its poof ) is based on an example

8 172 H.-J. Lai given by Jaege et al in [7]. Let m V 1 be an intege and let C 1 and C 2 be two disjoint cycles of 6m vetices. Denote C 1 ˆ u 1 u 2 u 6m u 1 and C 2 ˆ v 1 v 2 v 6m v 1 : Obtain a gaph J m fom the disjoint union of C 1 and C 2 by adding these new edges: 6 3m jˆ1 fu 2j 1v 2j 1 ; u 2j v 2j ; u 2j 1 v 2j ; u 2j v 2j 1 g. Note that J m has 3m disjoint K 4 's and evey independent set of J m has at most one vetex in each of these 3m K 4 's. Thus evey independent set in J m has at most 3m vetices. Note also that jv J m j ˆ 12m and je J m j ˆ 24m. It is easy to see that J m is 4-egula and 4-connected. We ague by contadiction to show that J m is not Z 3 -connected. Let b 1 1be a constant function de ned on V J m. Since jv J m j ˆ 12m, b A Z J m ; Z 3. Assume that thee is a function f A F J m ; Z 3 such that q f ˆ b. Expessing Z 3 ˆf0; 1; 1g and evesing the diection of edges in J m if necessay, we may assume that f e ˆ1; Ee A E J m. Since J m is 4-egula, each vetex in J m has eithe out degee 1 o 4. Let V i denote the numbe of vetices of out degee i in J m, whee i A f1; 4g. Then V 1 V 4 ˆjV J m j ˆ 12m and 4V 4 V 1 ˆ je J m j ˆ 24m. It follows that V 1 ˆ 8m and V 4 ˆ 4m. Howeve, since evey independent set in J m has at most 3m vetices, thee must be two vetices in V 4 that ae adjacent in J m, which is impossible since J m is 4-egula. Theoem 4.2 will follow fom the following stonge Theoem 4.7, which shows that the gaphs pesented in Example 4.3 ae basically the only exceptional gaphs fo a 3-edge-connected chodal gaph to be A-connected, fo any Abelian goup A with jaj V 3. We need two moe easy obsevations. Lemma 4.5. If G is simple, 2-connected, 3-egula and chodal, then G is isomophic to a K 4. Poof. Pick v A V G and denote N v ˆfv 1 ; v 2 ; v 3 g. Since G is 2-connected and simple, the two edges vv 1 and vv 2 must be in a shotest cycle C of G of length at least 3. Since G is chodal, C must have length exactly 3, and so v 1 v 2 A E G. Similaly, we may assume v 1 v 3 ; v 2 v 3 A E G. Since G is 3-egula and connected, V G ˆfv; v 1 ; v 2 ; v 3 g, and so G is isomophic to a K 4. Let K 4 be a given complete gaph on 4 vetices fu; v; x; yg with a distinguished edge a ˆ xy, and let G be a gaph disjoint fom this K 4 with je G j V 2 and with a distinguished edge a 0 ˆ x 0 y 0. De ne a new gaph G l K 4 to be the gaph obtained fom the disjoint union of G e 0 and K 4 by identifying x 0 and x to fom a new vetex, also called x, and by identifying y 0 and y to fom a new vetex, also called y. Note that the edge a ˆ xy is now an edge of G l K 4 and that G ˆ G l K 4 fu; vg. Lemma 4.6. G l K 4 AhZ 3 i if and only if G AhZ 3 i. Poof. Let G 0 ˆ G l K 4. We shall use the notation in the de nition of G l K 4 and

9 Goup Connectivity of 3-Edge-Connected Chodal Gaphs 173 assume that V G 0 ˆV G U fu; vg and V K 4 ˆfu; v; x; yg. In the poof below, K 4 denotes the given complete gaph on 4 vetices in the de nition of G l K 4. Assume that G AhZ 3 i. Let ˆfxu; xvg. Then G x; 0 Š has y as a cut vetex. Note that the gaph G x; 0 Š =G is spanned by a 3-cycle with one edge of this 3-cycle in a 2-cycle. By Poposition 3.2(C3) and by Lemma 3.3 (with n ˆ 2), G x; Š 0=G AhZ 3 i. By Poposition 3.2(C3) and by the assumption that G AhZ 3 i, G x; 0 Š AhZ 3i. By Lemma 3.1(i) and by G x; 0 Š AhZ 3i, G 0 AhZ 3 i. Convesely, assume that G 0 AhZ 3 i. Let b A Z G; Z 3. We shall show that thee is a function f A F G; Z 3 such that q f ˆ b. De ne b 0 : V G 0!Z 3 by ( b 0 z ˆ b z if z A V G 0 fu; vg 0 if z A fu; vg: Since V G 0 ˆV G U fu; vg and since b 0 u ˆb 0 v ˆ0, b 0 A Z G 0 ; Z 3. Since G 0 A Z 3, thee is a function f 0 A F G 0 ; Z 3 such that q f 0 ˆ b 0. Expessing Z 3 ˆf0; 1; 1g and evesing the diection on E K 4, we may assume that f 0 e ˆ1; Ee A E K 4. Since u and v has degee 3 in G 0, we may assume that the thee edges incident with u ae all oiented away fom u, and so the thee edges incident with v ae oiented into v. Let f be the estiction of f 0 on E G. Then it is easy to see that q f ˆ b. Since f 0 A F G 0 :Z 3, f A F G; Z 3. It follows by de nition that G AhZ 3 i. Remak. Lemma 4.6 povides an altenative poof fo the fact that the gaphs G m in Example 4.3 ae not Z 3 -connected. Theoem 4.7. Let G be a 3-edge-connected chodal gaph. Then one of the following holds: (i) G is A-connected, fo any Abelian goup A with jaj V 3; o (ii) G has a block isomophic to a K 4 ; o (iii) G has a subgaph G 1 such that G 1 BhZ 3 i and such that G ˆ G 1 l K 4. Poof. Let A be an Abelian goup with jaj V 3 and let G be a counteexample such that G BhAi and je G j minimized. Suppose st that G has a nontivial subgaph H (a subgaph with at least one edge) such that H AhAi. Then by the de nition of contaction, G=H, the gaph obtained fom G by contacting all edges in H, is also 3-edge-connected and chodal. Since je H j V 1 and since je G=H j ˆ je G j je H j, by the minimality of je G j, G=H A hai By Poposition 3.2(C3), G A hai, a contadiction. Theefoe, G does not have any nontivial subgaph H such that H AhAi. Since G is chodal, G must have a 3-cycle. By Lemma 3.3, if jaj V 4, then 3- cycles ae in hai. Since G must not have a nontivial subgaph in hai, it must be the case that A ˆ Z 3. By Lemma 3.3 and since G must not have nontivial subgaph in hz 3 i,we may assume that G has no loops and 2-cycles. Theefoe by the minimality of G and by Poposition 3.6, we assume that G does not satisfy any of (i)±(iii) of The-

10 174 H.-J. Lai oem 4.7, that G is simple, 2-connected and chodal, and that G does not contain a W 4 as a subgaph: 2 Let v A V G be an abitay vetex and let N v ˆfv 1 ; v 2 ;...; v m g. Since G is simple, m ˆ d v V 3. De ne N v ˆN v U fvg and let G v ˆ G N v Š and G v ˆ G N v Š be induced subgaphs of G. We st pove the following claims. Claim 1. G v is connected. Suppose that G v has moe than one components. Since G is 2-connected, thee is a shotest cycle C that contains v, one vetex in one component of G v and a vetex in anothe component of G v. Since G is a simple chodal gaph and since C is a shotest cycle, je C j ˆ 3 and so thee is an edge in G joining the two components of G v. Since G v is an induced subgaph, this edge should have been in G v, a contadiction. This poves Claim 1. Claim 2. Eithe G v is not 2-connected, o both d v ˆ3 and G v is isomophic to a K 4. Suppose that G v is 2-connected. If d v V 4, then by Lemma 4.1, G v must have a cycle of length at least 4. Since G is chodal, G v must have a cycle of length 4. Theefoe, G v has a W 4, contay to (2). Hence we assume that d v ˆ3. Then G v is a 2-connected gaph with 3 vetices. Hence G v is isomophic to a K 4. This poves Claim 2. Claim 3. Evey vetex v A V G such that G v is connected but not 2-connected must be in a vetex cut of cadinality 2 in G. Assume that v A V G is a vetex such that G v is not 2-connected and such that v is not in a vetex cut of cadinality 2. Let N v ˆfv 1 ; v 2 ;...; v m g with m ˆ d v V 3. Since G v is connected but not 2-connected, G v has a cut vetex v m (say). It follows that thee ae nontivial and connected subgaphs L 1 ; L 2 of G v such that G v ˆ L 1 U L 2 and such that V L 1 V V L 2 ˆfv m g. We may assume that v 1 is adjacent to v m in L 1 and v 2 is adjacent to v m in L 2. Since fv; v m g is not a vetex cut of G, G fv; v m g has a v 1 ; v 2 -path. Let P denote a shotest v i ; v j -path in G fv; v m g such that v i A V L 1 and v j A V L 2. Then since G is chodal, P must be a path of length one, and so v i v j A E G. It follows that v m is not a cut vetex of G v, a contadiction. This poves Claim 3. If G is cubic, then Theoem 4.7(ii) follows fom Lemma 4.5, contay to the assumption that G is a counteexample. Hence we may assume D G V 4. A subgaph H of G is called a 2-block if H is 2-connected and if ja G H j ˆ 2. A 2-block H of G is minimal if H is a 2-block and if H does not popely contain anothe 2-block of G. By Claim 1, Claim 2 and by the assumption that D G V 4, G must have a vetex v A V G such that G v is connected but not 2-connected. By Claim 3, G has a minimal 2-block H. Since H is 2-connected and simple, jv H j V 3. Since ja G H j ˆ 2, V H

11 Goup Connectivity of 3-Edge-Connected Chodal Gaphs 175 A G H 0q. Since H is a minimal 2-block, evey vetex in V H A G H cannot be in a vetex cut of cadinality 2. By Claim 1 and Claim 3, G v is 2-connected, Ev A V H A G H. By Claim 2, Ev A V H A G H, d v ˆ3 and G v is isomophic to a K 4. Thus if jv H j V 5, then one vetex in G v will be a cut-vetex of G, contay to the assumption that G is 2-connected. Hence jv H j ˆ 4 and so H is isomophic to a K 4. By Lemma 4.6, Theoem 4.7(iii) follows, contay to the assumption that G is a counteexample. This contadiction establishes the theoem. Note that K 4 BhZ 3 i. By Poposition 3.2(C2) and by Lemma 4.5, gaphs with a stuctue descibed in (ii) o (iii) of Theoem 4.7 cannot be in hz 3 i. Thus Theoem 4.7 can also be stated as the following chaacteization. Coollay 4.8. Let G be a 3-edge-connected chodal gaph. The G is A-connected fo evey Abelian goup A with jaj V 3 if and only if G does not have the stuctue descibed in Theoem 4.7(ii) and (iii). Poof of Theoem 4.2. When G is 4-edge-connected, neithe Theoem 4.7(ii) no Theoem 4.7(iii) will occu, and so by Coollay 4.8, G AhAi; EA with jaj V 3. Refeences 1. Aksionov, V.A.: Concening the extension of the 3-coloing of plana gaphs (in Russian). Disket. Analz. 26, 3±19 (1974) 2. Bondy J.A., Muty, U.S.R.: Gaph theoy with applications. Ameican Elsevie Catlin, P.A.: Gaph family closed unde contaction. Discete Math. to appea 4. GoÈtzsch, H.: Ein Deifabensatz fuè deikeisfeie Netze auf de Kugel, Wissenschaftliche Zeitschisch-Natuwissenschaftiche Reihe, 8, 109±120 (1958/1959) 5. GuÈbaum, B.: GoÈtzsch's theoem on 3-coloings, Mich. Math. J. 10, 303±310 (1963) 6. Jaege, F.: Nowhee-zeo ow poblems. In: L. Beineke et al.: Selected topics in gaph theoy, vol. 3. pp. 91±95. London New Yok: Academic Pess Jaege, F., Linial, N., Payan C., Tasi, M.: Goup connectivity of gaphs ± a nonhomogeneous analogue of nowhee-zeo ow popeties. J. Comb. Theoy, Se. B 56, 165±182 (1992) 8. Lai, H.-J.: Reduction towads collapsibility. In: Y. Alavi et al.: Gaph theoy, combinatoics, and applications. John Wiley and Sons pp. 661±670. (1995) 9. Lai, H.-J.: Extending patial nowhee zo 4- ows. J. Gaph Theoy, ±288 (1999) 10. Lai, H.-J., Zhang, C.-Q.: Nowhee-zeo 3- ows of highly connected gaphs, Discete Math. 110, 179±183 (1992) 11. Steinbeg, R., Younge, D.H.: GoÈtzsch's theoem fo the pojective plane. As Comb. 28, 15±31 (1989) 12. Steinbeg, R.: The state of the thee colo poblem. In: J. Gimbel et al.: Uuo vadis, gaphs theoy? Ann. Discete Math. 55, 211±248 (1993) 13. Thomassen, C.: GoÈtzsch's 3-colo theoem and its countepats fo the tous and the pojective plane. J. Comb. Theoy, Se. B 62, 268±297 (1994) 14. Tutte, W.T.: A contibution to the theoy of chomatic polynomials, Can. J. Math. 6, 80±91 (1954)

12 176 H.-J. Lai 15. Tutte, W.T.: On the imbedding of linea gaph into sufaces. Poc. Lond. Math. Soc., II Se. 51, 464±483 (1949) 16. Zhang, C.Q.: Intege ows and cycle coves of gaphs. New Yok: Macel Dekke 1997 Received: Januay 20, 1997 Revised: Novembe 16, 1998