SOM E RESUL TS ON R 2-ED GE-CONNECT IV ITY OF EVEN REGULAR GRAPHS

A pp l. M a th. J. Ch inese U n iv. S er. B 1999, 14 (3) : 3662370 SOM E RESUL TS ON R 2-ED GE-CONNECT IV ITY OF EVEN REGULAR GRAPHS Xu Junm ing Abstracṫ L et G be a connected k ( 3) 2regular graph w ith girth g. A set S of the edges in G is called an R 22edge2cut if G- S is disconnected and contains neither an iso lated vertex no r a one2 degree vertex. T he R 22edge2connectivity of G, deno ted by Κ (G ), is the m inim um cardinality over all R 22edge2cuts, w h ich is an impo rtant m easure fo r fault2to lerance of computer intercon2 nection netw o rk ṡ In th is paper, Κ (G ) = g (2k - 2) fo r any 2k2regular connected graph G ( K 5) that is either edge2transitive o r vertex2transitive and g 5 is given. 1 In troduction In th is paper, a graph G = (V, E ) alw ays m ean s a sim p le graph (w ithou t loop s and m u ltip le edges) w ith the vertex2set V and the edge2set E. W e fo llow [1 ] fo r graph2theo2 retical term ino logy and no tation no t defined here. It is w ell2know n that w hen the underlying topo logy of a com puter in terconnection net2 w o rk is m odeled by a graph G, the connectivity of G is an im po rtan t m easu re fo r fau lt2to l2 erance of the netw o rk. How ever, it has m any deficiencies ( see [ 2 ]). To com pen sate fo r the sho rtcom ings of the traditional connectivity m easu re, one can m ake u se of several gen2 eralized m easu res of connectednesṡ O ne of them is referred to as an R h2edge2conectivity p ropo sed by L atifi et al. [ 3 ]. L et G be a connected k2regu lar graph, k h+ 1. A set S of the edges in G is called an R h2edge2cu t if G - S is disconnected and con tain s neither an iso lated vertex no r a one2de2 gree vertex. T he R h2edge2connectivity of G, deno ted by Κ (h) (G), is the m in im um cardinali2 ty over all R h2edge2cu ts of G. O b serve that w hen h= 0, there w ill be no restriction on connected com ponen ts and w e have the traditional edge2connectivity. In addition, in the special case of h = 1, th is con2 nectivity w ill be reduced to the restricted edge2connectivity given in [2, 4 ]. T hu s th is con2 R eceived: 1998206223. R evised: 1998210212. 1991 M R Subject C lassification: 05C40, 68M 10. Keyw o rds: Connectivity, conditional connectivity, tran sitive graph ṡ T he w o rk w as suppo rted partially by NN SF of Ch ina (19671057).

X u J unm ing R 22ED GE2CONN ECT IV IT Y O F GRA PH S 367 nectivity can be regarded as a generalization of the traditional edge2connectivity, w h ich cou ld p rovide a m o re accu rate fau lt2to lerance m easu re of netw o rk s and has received m uch atten tion recen tly (fo r exam p le, see [2 6 ]). In th is paper w e restrict ou r atten tion to h= 2 and even regu lar graph ṡ Fo r the sake of conven ience, w e w rite Κ fo r Κ (2). G is called vertex2tran sitive if fo r any tw o vertices x and y in G, there is an elem en t Π # (G), the au tom o rph ism group of G, such that Π(x ) = y. It is w ell2know n that any ver2 tex2tran sitive graph is regu lar [ 7 ]. G is called edge2tran sitive if fo r any tw o edges e= xy and e = uv in G, there is an elem en t Ρ # (G) such that Ρ({x, y }) = {u, v}. Fo r a special class of vertex2tran sitive graph s referred to as circu lan t graph s, L i Q iao liang [ 5 ] their R 22edge2connectivity in h is Ph. D thesiṡ M o tivated by L i s w o rk, w e w ill, has ob tained in the p resen t paper, show that fo r a connected 2k2regu lar graph G ( K 5), Κ (G) = g (2k- 2) if G is either edge2tran sitive o r vertex2tran sitive and g 5. T he rest of th is paper is o rgan ized as fo llow ṡ tion s and p relim inary resu lts u sed in th is paper lateṙ in the p roofs of ou r m ain resu lts in 4. T he nex t section con tain s several no ta2 In 3, w e p resen t tw o lemm as u sed 2 Nota tion s and Prelim inary Results L et G be a k2regu lar graph. If k 2, then G certain ly con tain s a cycle. W e u se g (G ) to deno te the girth of G, the length of a sho rtest cycle in G. It is know n in [8, P rob lem 10. 11 ] that if G is a k2regu lar graph w ith girth g, then V (G) f (k, g ) = 1 + k + k (k - 1) +... + k (k - 1) (g - 3) g2, if g is odd; 2[1 + (k - 1) +... + (k - 1) (g- 2) g2 ], if g is even. A vertex x in G is called singu lar if it is of degree either zero o r one. L et X and Y be tw o disjo in t nonem p ty p roper sub sets of V. (X, Y ) = {e E (G) : there are x X and y Y such that e= xy E (G) }. If Y = X { = V gx, then w e w rite E (X ) fo r (X, X { ) and d (X ) fo r ge (X ) g. T he fo llow ing inequality is w ell2know n (see [8 ], P rob lem 6148). d (X Y ) + d (X Y ) d (X ) + d (Y ). (2) A n R 22edge2cu t S of G is called a Κ 2cu t if gs g= Κ (G) > 0. L et X be a p roper sub set of V. If E (X ) is a Κ 2cu t of G, then X is called a Κ 2fragm en t of G. fragm en t of G, then so is X { and bo th G [X ] and G [X { ] are connected. L et A Κ 2fragm en t X r (G) = m in{gx g: X is called a Κ 2atom of G if gx g= r (G). is a Κ 2fragm en t of G}. (1) It is clear that if X is a Κ 2 Since G [X ] is connected and con2 tain s no singu lar vertices fo r a given Κ 2atom X of G, G [X ] certain ly con tain s a cycle. T hu s r (G) = gx g g (G). Theorem 1. L et G be a connected 2k2regu lar graph, k 2. and Κ (G) g (2k- 2). If G K 5, then Κ (G ) ex ists Proof. L et G be a connected 2k2regu lar graph, G K 5 and k 2. W e w an t to show

368 A pp l. M a th. J. Ch inese U n iv. S er. B V o l. 14,N o. 3 Κ (G) g (2k- 2). Fo r th is purpo se, let X be the vertex2set of a sho rtest cycle C g in G. T hen X { g and E (X ) is an edge2cu t of G since k 2 and each in X in C g. If E (X ) is an R 22edge2cu t of G, then Κ (G) d (X ) = g (2k- 2). is a tw o2degree vertex Suppo se that E (X ) is no t an R 22edge2cu t of G. N o te the m in im ality of C g, it is clear that fo r any y X {, gn G (y ) X g 2 if g 4, by w h ich g = 3 and k= 2. L et y be a singu lar vertex in G - E (X ). T hen, obviou sly, y X { and y is a one2degree vertex in G - E (X ). L et Y = X {y }. T hen d (X ) = 6 and d (Y ) = 4. If there are no singu lar vertices in G - E (Y ), then E (Y ) is an R 22edge2cu t of G and Κ (G ) d (Y ) = 4< 6. Suppo se that there is som e singu lar vertex z in G- E (Y ). If z is an iso lated vertex in G - E (Y ), then G = K 5, w h ich con tradicts ou r assum p tion. T hu s z is a one2degree vertex in G- E (Y ). L et Z = Y {z }, then Z θ g, d (Z ) = 2 and G - E (Z ) con tain s no singu lar verticeṡ It fo llow s that E (Z ) is an R 22edge2cu t of G. So Κ (G) d (Z ) = 2< 6. 3 Two L emma s L emma 1. L et G be a connected 2k2regu lar graph, G K 5 and k 2. L et R be a p roper sub set of V (G ) and U be the set of the singu lar vertices in G - E (R ). d (R ) Κ (G) + 1, then gr g< g (G). Proof. L et g = g (G). If g U Α R and Since Κ (G) ex ists, Κ (G ) g (2k - 2) by T heo rem 1. Suppo se to the con trary that gr g g. W e w an t to derive con tradiction ṡ If G [R ] con tain s no cycles, then ge (G [R ]) g gr g- 1. So w e can deduce a con tradic2 tion as fo llow ṡ g (2k - 2) + 1 Κ (G) + 1 d (R ) = 2kgR g - 2gE (G [R ]) g 2kgR g - 2 (gr g - 1) = gr g (2k - 2) + 2 g (2k - 2) + 2. If G [R ] con tain s cycles, then let R be the vertex2set of the un ion of all b lock s that con tain a cycle in G [R ]. T hu s U Α R gr. N o te that gn G (u) R g 1 fo r any u R gr and k 2, G - E (R ) con tain s no singu lar verticeṡ T h is im p lies that E (R ) is an R 22edge2cu t of G, by w h ich d (R ) Κ (G). By the cho ice of R w e have that fo r any tw o distinct ver2 tices in R, their neighbo rs in R gr are disconnected in G [R ] and that fo r any neighbo r z of R in R gr, either z U o r there is a path in G [R gr ] connect ing z to som e vertex in U. T hu s g (R gr, R ) g gu g, and g (R gr, R ϖ ) g gu g (2k- 1) since U R gr. W e can de2 duce a con tradiction as fo llow ṡ Κ (G) d (R ) = d (R ) - g (R gr, R ϖ ) g + g (R, R gr ) g T he p roof is com p lete. d (R ) - gu g (2k - 1) + gu g = d (R ) - gu g (2k - 2) Κ (G) - 1. L emma 2. L et G be a connected 2k2regu lar graph. fo r any tw o distinct Κ 2atom s X and X of G. If Κ (G) < g (2k- 2), then X X = g Proof. Suppo se that Κ (G ) < g (2k - 2) and X and X are tw o distinct Κ 2atom s of G.

X u J unm ing R 22ED GE2CONN ECT IV IT Y O F GRA PH S 369 N o te that gx g = gx g = r (G ) g. If r (G ) = g, then G [X ] is a cycle of length g. T hu s g (2k- 2) = d (X ) = Κ (G) < g (2k - 2). T h is con tradiction im p lies that gx g > g. W e w an t to show that X X = g. Suppo se to the con trary that X X g. L et A = X X, B = X X, C = X { X, D = X { X. T hen gd g ga g 1, gb g= gc g= r (G) - ga g 1 since X and X are tw o distinct Κ 2atom s of G. To derive con tradiction s, w e con sider tw o cases separately. Ca se 1 G- E (A ) con tain s no singu lar vertices. It is clear that E (A ) is an R 22edge2cu t of G and G [A ] certain ly con tain s cycles since G - E (A ) does no t con tain any singu lar vertex. It fo llow s that d (X X ) = d (A ) > Κ (G), gd g ga g g. (3) N o ting d (X ) = d (X ) = Κ (G), by (2) and the left inequality in (3), w e have d (D ) = d (X X ) d (X ) + d (X ) - d (X X ) < Κ (G). T h is im p lies that G - E (D ) does certain ly con tain som e singu lar vertices o therw ise E (D ) is an R 22edge2cu t of G w ho se cardinality is less than Κ (G ). T hese singu lar vertices are con tained in D obviou sly. T hu s gd g< g by L emm a 1. T h is con tradicts (3). Ca se 2 G- E (A ) con tain s singu lar verticeṡ L et y be a singu lar vertex in G- E (A ), then y A obviou sly. L et Y = X g{y } if g (y, C ) g> g (y,b ) g (o r let Y = X g{y } if g (y, C ) g< g (y, B ) g), then gy g= gx g- 1 and d (Y ) d (X ) - g (y, D ) g - g (y, C ) g + g (y, B ) g + 1 d (X ) = Κ (G). (4) N o te that X then they all are con tained in Y. is a Κ 2atom of G and Y < X, there ex ist singu lar vertices in G - E (Y ), T h is con tradicts the fact that gx g> g. so g Y g < g by L emm a 1, by w h ich gx g = g Y g + 1 g. W e can sim ilarly ob tain a con tradiction if w e con sider the case of g (y, C ) g< g (y, B ) g. N ex t, w e w an t to con sider the case of g (y, C ) g = g (y, B ) g. N o te that in th is case the e2 quality in (4) does no t ho ld on ly w hen g (y, D ) g = 0 and y is a one2degree vertex in G - E (A ). It fo llow s that d G (y ) = 1+ g (y, C ) g+ g (y, B ) g, w h ich is odd. T h is con tradicts ou r assum p tion that the regu larity of G is even. T he p roof of L emm a 2 is com p lete. 4 M a in Results Theorem 2. L et G be a connected 2k2regu lar edge2tran sitive graph, G K 5 and k 2, then Κ (G) = g (2k- 2). Proof. By ou r assum p tion, Κ (G) ex ists and Κ (G) g (2k - 2) by T heo rem 1. Suppo se that Κ (G) < g (2k- 2). L et X be a Κ 2atom of G, then gx g > g 3. L et e= xy be an edge in G [X ] and e = y z be an edge in E (X ), z X {. Since G is edge2tran sitive, there is Ρ # (G) such that Ρ({x, y }) = {y, z }. H ence Ρ(X ) is also a Κ g gatom of G. L et X = Ρ(X ), then X X since z X and z X. O n the o ther hand, since y X X, X = X by L em 2

370 A pp l. M a th. J. Ch inese U n iv. S er. B V o l. 14,N o. 3 m a 2. T h is con tradiction im p lies that T heo rem 2 ho ldṡ Theorem 3. L et G be a connected 2k2regu lar vertex2tran sitive graph, g 5 and k 2, then Κ (G) = g (2k- 2). Proof. It is clear that Κ (G ) ex ists and Κ (G ) g (2k - 2) by T heo rem 1. Suppo se that Κ (G) < g (2k - 2) and X the end w e let is a Κ 2atom of G. W e claim that G [X ] is vertex2tran sitive. 0 = {Π # (G) : Π(X ) = X }, 7 = {Π 0 : x X ] Π(x ) = x }. It is clear from L emm a 2 that 0 is a subgroup of # (G), and the con stituen t of 0 on X acts tran sitively and 7 is a no rm al subgroup of 0. T hu s there is an in jective hom om o rph ism from the quo tien t group 0 g7 to # (G [X ]) w here by each co set of 7 is associated w ith the restriction to X of any rep resen tative. L et the regu larity of G [X ] be t, then 2 t 2k- 1 and g (2k - T h is p roves that G [X ] is vertex2tran sitive. 2) > Κ (G) = d (X ) = (2k - t) gx g. (5) Since G [X ] is t2regu lar and t 2, G [X ] certain ly con tain s a cycle of length at least g. fo llow s from (1) and (5) tha t 0 < g (2k - 2) - (2k - t) f (t, g ). (6) T he righ t side of (6) is an increasing function w ith respect to t and is a descending function w ith respect to g. It is no t difficu lt to show that there ex ists no t [ 2, 2k - 1 ] such that (6) ho lds if g 5. T h is p roves T heo rem 3. To It References 1 Bondy, J. A., M urty,u. S. R., Graph T heo ry w ith A pp lications, M acm illan P ress, L ondon, 1976. 2 E sfahanian, A. H., Generalized m easures of fault to lerance w ith app lication to N 2cube netw o rk s, IEEE T ranṡ Compuṫ, 1989, 38: 1586 1591. 3 L atifi, S., H egde,m., N aragh i2pour,m., Conditional connectivity m easures fo r large m ultip rocesso r system s, IEEE T ranṡ Compuṫ, 1994, 43: 218 221. 4 E sfahanian,a. H., H ak im i, S. L., O n computing a conditional edge2connectivity of a graph, Info rm a2 tion P rocessing L etters, 1988, 27: 195 199. 5 L i,q. L., Graph theo retical studies on fault2to lerance and reliability of netw o rk s (Ch inese): [Ph. D. T hesis],u niversity of Science and T echno logy of Ch ina, H efei, 1997. 6 L i,q. and Zhang, Y., R estricted connectivity and restricted fault diam eter of som e interconnection net2 w o rk s,d IM A CS Ser. D iscrete M ath. T heo reṫ Compuṫ Sci., 1995, 21: 267 273. 7 W atk ins,a. E., Connectivity of transitive graph s, J. Com bin. T heo ry, 1970, 8: 23 29. 8 L ovasz, L., Com binato rial P roblem s and Exercises, N o rth2ho lland Publish ing Company, Am sterdam, N ew Yo rk,o xfo rd, 1979. D ep ṫ of M ath., U niv. of Science and T echno logy of Ch ina, H efei 230026. Em ail: XU JM @U STC. EDU. CN