All other standard graph-theoretic terms that are used in this paper can be found for instance in 13] .

Vertex splitting, parity subgraphs Definition 1.2 I~et G be a bridge-less groph. A family C of circuits of G and circuit covers is called a circuit cover of G iJ cach edge of G is contained in some circuit ofC. Cun-Quan Zhang' Definition 1.3 Let C be a circuit cover of a graph G and v E V(G). The Department of Mathematics depth of the circuit cover C at the vertex v i., the number of circuits ofC West Virginia University containing v and is denoted by cdc(v). The depth of the circuit cover C is Morgantown, West Virginia 26506-6310 max{cdc(v) : v E V(G)} and is denoted by cdc(G). email: [email protected] The following (!onjecture was presented by Laszl

Conject~re 1.1 (Pyber, If)) Each bridge-less glnph G with maximum degree .6.( G) has a circuit cover C such that AUSTRACT. Let G be a 2-edge-coTlnccted graph and v be a vertex of G and F C F' c E(v) such that 1 ~"IFI and cdc(G) ::; .6.(G). IFI + 2 = W/I ~ d(v) - 1. Then there is a subset P* such 1Iiat F C F* C F' (here, IF' I = IFI + I), and the graph ob­ In this paper, we prove that tained from G by splitting the edges of F* away from v remains 2-edge-connectcd unless v is a cut-vcrtex of G. This generalizes Theorem 1.1 Every blidge-Iess groph G has a circuit cover C such that a very useful Vertex-Splitting Lemma of Flcischncr, Let C be G) a circuit covcr of a bridge-less graph G. The depth of C is the the depth of C is at most .6.( + 2. smallest intcgcr k such that every vertex of G is contained in at most k circuits of C. It is conjectured by L. Pyber that every Theorem 1.2 If a bridge-less graph G admits a nowhere-zero 4-flow or bridge-less graph G have a circuit cover C such that the depth contains no subdivision of the Peter.~en gmph, then G has (l circuit cover C of C is at 1Il0st tl.(G). In this papcr, we prove that (i) . every such Utat the depth ofC is at most 2r.6.(G)j3]. bridge-less graph G has a circuit cover C such that the depth of Cis at most tl.(G) +2 and (ii). if a bridge-less graph G admits a Let II be a subgraph of G. The set of edges of II incident with v is nowhere-zero 4-now or contains no subdivision of the Petersen denoted by E/I (v). The number of components of a graph G is denoted by graph, then G has a circuit cover C such that the depth of C is w(G). at most 2r tl.(G)/3l- Definition 1.4 Let G be a gmph and v be a vel'tex of G and F c E(v). The gmph Glu;F! obtained from G by splitting the edges of F away from v, (that is, adding a Tlew vertex v' and changing Ule end v of the edges of F 1 Introduction to be v'. See fi,qure J). Note that the new vertex cr'eat ed in the .~pliltin,q F Definitioll 1.1 A circuit oj a graph is a connected f-regular subgroph, away fmm v i., always denoted by v' in the contest. while a cycle oj a grnph i., a .,ubgraph with even degree at every vertex. A bridge b of a groph G i., an edge of G that is not contained in any circuit ofG. Theorem 1.3 (Fleischner (4), or see 17, 8)) Let G be a connected and bridge-less graph and v E V(G) (with d(v) ~ 4) and eO,elte2 E Ec(v). ·Partial support provided by the National Science Foundation under Grant DMS- 9306379 In the case of v being a cut-vertex, choose co, C2 from distinct blocks of C. Then either Glu;{eo,c,}! or Glu;{Co.c,}! is connected and bridge-Ie.,s.

JCMCC 20 (1996), pp. 206-216 207 if either v' and v are in two distinct components or there ar'e a pair of edge-disjoint paths in G(v;FJ joining v' and v.

Proof. If G(v;F] has a bridge b, then there must be a path P of G(v;FJ 0----0 0----0 joining v', v and containing b since G is bridge-less and each circuit of G containing b is broken into a path because of the splitting of the vertex v. Thus, v', v are in the same component of G(v;FJ and we assume that there are a pair of edge-disjoint paths S I, S2 joining v', v in G(v;FJ. Without loss of generality, let SI be a path not containing the edge b. Thus, the syrrunetric difference of P and SI yields a cycle of G(v;F] containing b. This contradicts that b is a bridge of G(v;F]' The another direction of the lemma is tri vial. 0

Lemma 2.2 Let G be a connected, bridge-le.~s graph and v be a ver·tex of G and F c.E(v) such that 2 ~ IFI ~ dc(v) - 2. If the graph G' = G(v;F( has a br'idge C E Ec'(v' ) (or e E Ec'(v)), then either G(v;F\{cl] or G(v;FU{cl] is F not connected. 0- o :.- F -0 Proof. Without loss of generality, we supposc that G is connected. Assume E Ec'(v ). v'x Figure 1: G and Glv;F]: splitting F away from v that G' has a bridge e ' The bridge e = must be contained in a path connecting v and v' since G is bridge-less and connect.ed. lIence G' is connected and G' \ {c} is discolHlccled with x contained in the component Theorem 1.3 is the well-known vertex-splitting lemma which has been containing v but not v'. Therefore G(v;F\{e}J is disconnected. 0 used by various authors in the studios of compatible decompositions, integer Proof of Theorem 1.4. Let F' \ F = {CI,C2} ' Let lIj = G(v;I-'U{c;}] for flows, cyclc covers and graph colorings. (For instance, [4], [5], [6], [7], [8], j = 1,2. As.'iume that both III and 112 have bridges and hoth G(v;FJ and [9], [12J, [1:3J, [14], [16], [19J. [20]' etc.) The following theorem is an analogy G(v;F'] are connected. of Thwrcrn 1.3 which can be used for splitting a graph to meet certain By Lemma 2.2, neither el nor e2 is a bridge of Hj (j = 1,2) since both dcgn-,e requirement. G(v;FJ and G(v;l"'] are connected. By Lemma 2.1, 1I1 doC'." not have a pair of edge-disjoint paths joining v and v'. lienee, CI, C2 arc in two distinct Theorem 1.4 Let G be a connected and bridge-less graph and v be a vertex blocks, say [(I and [(2, of III. Since neither CI nor e2 is a bridge of HI, ofG and Fe F' c E(v) such that 1 ~ IFI and IFI + 2 = IF'I ~ d(v)-1. 1<1,1<2 are non-trivial bJocks. Let Ci be a circuit of [(i containing ei for Then there is a subset F* .meh that F C F* C F' (he7'e, IF'I = IFI + 1). and i = 1,2. In the graph }[2 = G(v;Fu{e.1J' the circuits C I ,C2 are broken G(v;FOI i.~ connected and bridge-less, unle.~s G(v;FJ or G(v;F'] is disconnected. into two edge-disjoint paths connecting v and v'. By Lemma 2.1, H2 is bridge-less. 0 Thcorcm 1.4 and ThcorcHl 1.:1 have the salllc property that the vertex­ Now, we are to deduce Theorem 1.3 from Theorem 1.'1. splitting operation prescrves thc embedding property of graphs. Proof of Thcorcm 1.3. It is obvious if v is not a cut-vertex of G. Assume that v is a cut-vertex of G. Let F = {eo} and F' = {co, el, e2}. If G(v;FI 2 Vertex splitting is disconnected then CO is a bridge of G. This contradicts that G is bridge­ A few lemmas are needed before the proof of Theorem 1.4. less. Now, by Theorem 1.4, we only need to show that G(v,F'1 is connected. Note that each non-trivial block of G contains at least two edges and G is Lemma 2.1 Let G be a b7idge-less graph and v be a vertex of G and F C bridge-less, for i = 0,2 let J; (Ii # cd be an edge of Ec(v) contained in the block of G cont.aining ei. Since eo, C2 are chosen from distinct blocks of E(v) .,ueh that 2 ~ IFI ~ d( v) - 2. Then G(v;FI i., bridge-less if and only G and I{eo, e2.!o, h}1 = 4 and lF'l = 3, EC1v:F'1 (v) contains at least one of

208 209 {fo,h} and hence, v',v are in the same component of Gjv;F'j. Therefore, and Gjv;F'j is connected and Theorem 1.3 follows. 0 0 if de (v) == 0 mod 3 s(v) = 1 if dc(v) == 2 mod 3 Remarks. { 2 if de(v) 1 mod 3. Similar to Theorem 1.3, the splitting operation induced in Theorem 1.4 == also preserve the embedding property of G. Let the edges incident with a Definition 3.3 A subgraph P of a graph G is a parity subgraph of G if vertex v be arranged on a surface in the order as el, ' " ,ed. If one chooses dp(v) == de(v), (mod 2) for each vertex v E V(G). F' = {ej,eHI,'" ,ej_l,ej} and F = {eHI,'" ,ej-d, then the graph Gjv,F'j preserves the embedding property on the sarne surface where F* is Theorem 3.2 Det G be a graph. Then G has a parity .,ubgraph P such one of {ej,ei+l, ... ,ej-d and {ei+l,'" ,ej}' that for each vertex v E V(C) Note that the vertex splitting operations in Theorem 1.3 and its general­ ization, Theorem 1.4 preserve the 2-edge-connectivity of graphs. While for dc (v)/3 if dc(v) == 0 mod 3 higher edge-connectivity, Nash-Williams has the following theorem. (Note, dp(v) ~ (dc(v) - 2)/3 + 2 if dc(v) == 2 mod 3 { the following theorem does not preserve the embedding property of graphs) (dc(v) - 4)/3 +" if dcCv) == 1 mod 3.

Theorem 2.3 (Na .~h- Williams [22J) Let k be an even integer and G be a Proof. By Lemma 3.1, let n be a vertex splitting of G such that neG) is k-eonnccted gmph and v E V(G). Let a be an integer such that k ~ a ~ described in Lemma 3.1. The underlying graph of 1I'(G) is cubic and bridge­ d( v) - k. Then there is an edge sub.~et F C E( v) .~uch that IFI = a and less and therefore has a perfeEt matching M, (Petersen Theorem [17J, or Gjv;FJ is k-edge-connected. see [3J p79). The subgraph Q of neG) induced by the edges of M is a parity subgraph of 1I'(G) such that 3 Parity subgraphs d (v) = { 1 if d,,(c)(v) = 3 Definition 3.1 Let G and II be two gmphs. The graph II is said to be Q 0 or 2 if d,,(c)(v) = 2. obtained from G by a vertex splitting n if V(H) has a partition {U, : i = 1" " ,11} such that G can be obtained from IJ by identifying every Uj Thus the subgraph P of G induced by the edges of E(Q) is a parity subgraph (i = 1"" , n) to a single vertex Vi ' If a gmph IJ is obtained from G by of G satisfying the description of the theorem. 0 a vertex splitting n, then we write II = 1I'(G) and Ui = neVi) for each The properties of parity suhgrnphs and their relations with the prob­ Vi E V(G). (lIere, we simply consider E(G) = E(n(G»). lems of integer flow, circuit cover can be found in the papers [15, 25, 26J. Theorem 3.2 will be applied in next section. Definition 3.2 A graph G is quasi-cubic if the degree -of each vertex of G is either- three or two. 4 Depth of circuit cover By recursively applying Theorem 1.4 to each vertex of a graph with 4.1 A general upper bound degree at least four (with a, b = 3, or 2), we have the following corollary, The following theorem (the 8-Flow Theorem) is one of the fundamental Lemma 3.1 Det G be a bridge-less graph. Then G has a vertex-splitting 11' results that. we will use in this paper. such that {1}. n( G) is bridge-less; Lemma 4.1 (Jaeger [llJ, or sec [13J) Every bridge-less groph G has a (2). 1I'(G) is quasi-cubic; cycle cover consisting of at mo.~t three cycles. (3). For each vertex v E V(G) with degree dcCv) ~ 2, 1I'(v) is a set oft(v) degrce-three-vertices and s( v) degree-two-vertices where Note that a cycle is a union of edge-disjoint circuits. As an immediate corollary of Lemma 4.1, the following result was originally observed by dc(v)/3 if dc(v) == 0 mod 3 Pyber ([2]). t(v) = (dc(v) - 2)/3 if de(v) == 2 mod 3 { (dcCv) - 4)/3 if de(v) == 1 mod 3,

210 211 Theorem 4.2 Each bridge-less gmph G WiUl maximum degree t:.(G) has This result is to be generali7.ed in this section. The following lemmas are a circuit cover C such that fundamental in this section. 3 cdc(G) ::; 2t:.(G). Lemma 4.7 (Jaeger [11/, or see [13/) Every 4-edge-connected groph G admits a nowhere-zero 4-flow. However, for a quasi-cubic graph, Theorem 1.2 can be improvcd Lemma 4.8 (Zhang {24/, or .'ee (26, 10/) Let G be a groph and P be a Lemma 4.3 Each bridge-less quasi-cubic groph G has a circuit cover C parity .mbgraph oj G. IJ G admits a nowhere-zero 4-flow, then G has a such Uwt circuit cover C such that each edge e E E(P) i., contained in precisely two cdc(G) ::; 3. circuit., oj C and eaeh edge e E E( G) \ E(P) is contained in precisely one ciJ'cuit oJC. Proof. For!l. quasi-cubic graph, each cycle of G is the union of vertex­ disjoint circuits. Thus, by Lemma 1.1, for a 3-cycle cover of G, each vertex Lemma 4.9 (Alspach, Goddyn and Zhang /1/, or .~ee (IO, 25/) Let G be is contained in at mo.c;t three circuiLc;. 0 a bridge-le.qs gmph and P be a parity s1,bgrop" of G . If G contains no An ilTlIlleciiate corollary of Lelllma 3.1 and Lelllma 1.:1 is the following subdivi.9ion oj the PeteJ·.gen g/Uph, then G has a circnit cover C .9uch that result which generalizes Theorem 1.2. each edge e E E(P) is contained in precisely two circuits ofC and each edge e E E(G) \ E(P) is contained in precisely one circuit ofC. Theorem 4.4 Let G be a bridge-les.~ graph. Then G has a circuit cover C such that Jar each vedex v E V (G) (Please refer to [13] or [23] for the definition and properties of integer flows, and refer to [15] and [26] for the relations of parity subgraphs and dc(v) iJ dc(v) == 0 lIIod 3 integer nows, circuit covers.) cJc(v) ::; dc(v) - 2 + 3 = dc(v) + 1 iJ dc(v) == 2 mod 3 { Now we are ready to usc Theorem 3.2 and Lemmas 1.7, 4.8, 1.9 to obtain dc(v) -1 + 6 = dc(v) + 2 iJ dc(v) == I lIIod 3 the following theorems.

The following result and Theorem 1.1 are immediate corollaries of The­ Theorem 4.10 Let G be a bridge-less groph. IJG admits a nowhere-zero 4- orem 1.1. flow or contains no subdivision oj the Petersen graph or is ../ -edge-connectcJ., then G has a circuit cover C ,mch that Corollary 4.5 /,et G be a bridge-les.~ graph with the maximum degree t:.(G). Then G has a circuit cover C such that ldc(v) iJ dcev) == 0 mod 3 cdcev) ::; ~(dG(v) + 1) iJ dc(v) == 2 mod 3 t:.(G) iJ t:.( G) == 0 mod 3 { 3(dc (v) + 2) if dc(v) == 1 mod 3 cdc(G) ::; t:.(G) + 1 iJ t:.( G) == 2 mod 3 { t:.(G) + 2 iJ t:.(G) == 1 mod 3 Proof. Let P be a parity subgraph of G described in Theorem 3.2. Let C be a circuit cover of G covering each edge of P twice and each edge of 4.2 A bettcr bound for ccrtain families of graphs E(G) \ E(P) once. Since a circuit is a connected 2-regular subgraph, for each vertex v E V(G), the number of circuits of C containing v is It was ohserved by Pyber (2) that the famous Circuit Double Cover Conjec­ ture (Conjecture 5.1 in next section, hy Szekeres [21], Seymour [18]) implies 1 1 -[2<11'(11) + (dc(v) - dl'(v)] = -(dc(v) + d/,(v)) Conjecture 1.1. Since we have known ([12], [1]) that a graph admitting a 2 2 nowhere-7.ero 1-now or containing no subdivision of the Petersen graph has !(dc(v) ~dc(v)) = ~dc(v) a circuit double cover, Conjecture 1.1 holds for those graphs. That is, + if dc(v) == 0 mod 3 1[dc (v) + ~(dc(v) - 2) + 2] = ~(riC(lI) + 1) if dc(v) == 2 mod 3 Theorem 4.6 1J a bridge-les.~ graph G admits a nowhere-zero 4-flow or 2[dc (v) + 3(dc(v) - 4) + 1J = 3(dc(v) + 2) if dc(v) == 1 mod 3 contain., no subdivision oj the Petersen gmph, then G has a circuit cover C o such that cdc(G) ::; t:.(G). An immediate corollary of Theorem 4.10 is Theorem l.2.

212 213 5 Edgo-depth of circuit cover References Definition 5.1 Let C be a circuit cover of a graph G and e E E(G). The [1] n. Alspach, L. Goddyn, and C.-Q. Zhang, Graphs with the circuit edge-depth of the circuit cover C at the edge e is the number of circuits cover property, 1Tansaclion AMS (to appear) of C containing e and is denoted by cede (e). The edge-depth of the circuit cover Cis max{cede(e) : e E E(G)} and is denoted by cede (G). [2] J. Bang-Jensen and B. Toft, Unsolved problems presented at the Julius Petersen Graph Theory Conference, Discrete Mathematics, 101 The following conjecture is an equivalent version of the famous Circuit (1992),351- 360. Double Cover Conjecture. The problem of circuit double cover has been [3] J. A. Bondy and U. S. Il. Murty, Graph Theory with Applications, extensively studied in recent years. (See surveys, [10, 12, 13, 25], etc.) Macmillan, London, 1976.

Conjecture 5.1 (Szekere., f21J, Seymour f1B]) Every bridge-Ies., grnph G [1] II. Fleischner, Eine gerneinsame Basis fiir die Theorie der eulerschen has a circuit cover C such that cedc(G) ~ 2. Graphen und den Satz von Petersen, Monatsh. Math. 81 (1976), 267- 278. Note, the 8-now theorem (Lemma <1 .1) implies that every bridge-less graph has a circuit cover with the edge-depth at most three. [5] H. Flcischner, Eulerian Graph, in Selected Topics in Graph Theory 2 (L. Bcineke and R. Wilson, cds.), Wiley, New York, 1983, 17 -53. Definition 5.2 Let G be a bridge-less grnph. A ci7'cuil cover C is shortest [6] II. Fleischner, Some blood, rsweat, but no tears in eulerian graph theory, if the total length of circuits of C is shoTtest among all cir'cuit covers of G. Congressus Numemntiu11I, 63 Nov. (1988), 9-48.

There arc many arlicles in the topic of shortest circui t cover. (See survey, [7J II. Fleischller and M. Fulrnek, P(D)-CompaLible Eulerian Trails ill Di­ [10]' etc.) graphs and a New Splitting Lemma, in Contemporary MeU/Ods in Graph Theory (edited uy It Bodendiek), (1990), 291 - 303. Conjecture 5.2 (Zhang f25J) Every b7idge-less gmph G has a shortest circuit cover C with cede (G) ~ 2. [8] II. Fleischner, Eulerian Graphs and Related Topics, Part 1, Vol. 1, Annal., of Discrete Mathematics, Vol. 15, North-Holland, 1990. Conjecture 5.2 implies Circuit Double Cover Conjecture (Conjecture 5.1). [9] D. A. IIolLon and .I. Sheehan, The Pet:ersen Graph, Aust.ralian Math­ Conjecture 5.2 holds for certain families of graphs, for example, the graphs ematical Society Lecture Smies 7, Cambridge University Press, 1993. admitting nowhere-zero 4-now (Lemma 4.8) and the bridge-less graphs con­ taining no subdivision of the Petersen graph (Lemma 4.9). On the other [10] 13. Jackson, On circuit covers, circuit decompositions and Euler tours hand, we do not even know if there exists a constant upper bound for of graphs, British Combinatorics Conference, preprint (1!)93). the edge-depths of shortest circuit covers for all bridge-less graphs. The following is a weak version of Conjecture 5.2. [11] F. Jaeger, Flows and generalized coloring Lheorems in graphs, J. Com­ binatorial Theory B, 26 (1979), 205- 216. Conjecture 5.3 There is an integer K such that every bridge-less graph [12] F. Jaeger, A survey of the cycle double cover conjecture, in Cycles in G has a shortest circuit cover C with cede (G) ~ IC Graphs (13. Alspach and C. Godsil, cck), Annals of Discrete Mathe­ matic.', 27 (1985), 1-·12.

[13] F. Jaeger, Nowhere-zero now problems, in Selected Topics in Graph Thcory 3 (L. Beineke and Il. Wilson, cds.), Wiley, New York, 1988, 71-95.

[1t1] T. It Jensen, nIUe's k-f1ow problems, Master's Thesis, Ordense Uni­ versity, 1985

214 215 (I 5) H.-J. Lai and C.-Q. Zhang, Nowhere-zero 3-flows of highly connected graphs, Discrete Mathematics 110 (1992), 179-183.

(16) M. Moller, H. G. Carstens and G. Brinkmann, Nowhere-zero flows in On finite bases for some PBD-closed sets low genus graphs, J. Gmph Theory 12 (1988), 183-190.

(17) J. Petersen, Die Theorie der reguJiiren graphs, Acta Math. 15 (1891), B. Ou 193-220. Dcpartmcnt of Mathemati(~<; (18) P. O. Seymour, Sums of circuits, in Graph Theory and Related Topics Suzhou University (.I. A. Bondy and U. S. R. Murty, eds.), Academic Press, New York, Suzhou 215006 (1979), 342-355. P.R. of China (19) P. D. Seymour, Nowhere-zero 6-flows, J. Combinatorial Theory, B 30 (1981),130-135.

(20) R. Steinberg, Tutte's 5-flow conjecture for projective plane, J. Graph 1 - ABSTRACT - {1i' 0 ( . Let 1/a - . v>_a,v= ,1 mod a)}. It is well Theory 8 (1984), 277-285. known that such sets are PBD-closed I~' 't b f, h . Illi e ases are found [21] G. Szekcres, Polyhedral decompositions of cubic graphs, J. Austral. thor t ese sets for a = (; ,.7 and 8 At the s arne t'line we .Improve Math. Soc. 8 (1973), 367-387. e rcsu~ of Mullin in [6J about finite bases of 1/(1 = {v: v > a+l,v=! (mod a)} fora=5and6. - [22J C. St. J. A. Nash-Williams, Connected detachments of graphs and generalized Euler trials, J. London Math., Soc., (2) 31 (1985), 17-29.

(23) O. H. Younger, Integer flows, J. Graph Theory 7 (1983), 349-357. 1 Introduction

The terrninology and n l' t' l' [24] C.-Q. Zhang, Minimum cycle coverings and integer flows, J. Graph o a Jon on lms paper follow from that of [6,8]. Let

Theory 14 (1990), 537-546. fl H ={v:v2:a+l,v=1 (mod a)} [25] C.-Q. Zhang, Cycle cover theorems and their applications, in Graph H~ = {v: v 2: a,v =0, 1 (mod a)}. Structure Theory (cds. N. Robertson and P. Seymour), Contempomry Mathematics 147 (1993), 405-418. It is well known that such sets are PBD-c1osed F"l b h found for lI a when 2 < < ( . Inl cases ave been [8]). - a - 7 see [6,8]), and for J1~ when 2::; a ::; 5 (see [26] C.-Q. Zhang, Nowhere-zero 4-flows and cycle double covers, Discrete Math., (to appear). In t~is pape~, finite bases are found for II~ for a = 6, 7 and 8 ;;:: ~J~:n~e61.mprove the result of Mullin in [6] about finite bas~s ~~ ~~:

For our proof, we record the well-kn b . plete TD (or lTD) TD(k n)-TD(k )~n 0 servatlOn below. An incorn- satisfies the following pro'perties: ,m IS a quadruple (X, G, H, A), whieh

(1) X is a set of cardinality kn,

(2) G = {G;: 1 ::; i ::; k} is a partition of X into k groups of size n,

(3) H = {II;: 1::; i::; k}, where each G· ~ H d IIII . , " an i = m, I ::; l ::; k,

216 JCMCC 20 (1996), pp. 217-224