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Vertex Splitting, Parity Subgraphs and Circuit Covers

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Vertex Splitting, Parity Subgraphs and Circuit Covers 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<l Pyllcr a.t the Julius Petcrsen Graph Theory Conference, Denmark, l!J90. 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.
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