CLONES IN 3-CONNECTED FRAME MATROIDS

Transcription

1 CLONES IN 3-CONNECTED FRAME MATROIDS JAKAYLA ROBBINS, DANIEL SLILATY, AND XIANGQIAN ZHOU Abstract. W dtrmin th structur o clonal classs o 3-connctd ram matroids in trms o th structur o biasd graphs. Robbins has conjcturd that a 3-connctd nonuniorm matroid with a clonal class o siz q 1 is not GF (q)-rprsntabl. W conirm th conjctur or th class o ram matroids. 1. Introduction W will assum that th radr is amiliar with matroid thory as prsntd by Oxly in [4]. Th graph-thory trminology usd is mostly standard. A ram matroid is a matroid M which has an xtnsion M J satisying th ollowing. r(m) = r(m J ) E(M J ) = E(M) J and J is a basis o M J. For vry nonloop E(M) that is not paralll to som b J, thr is a uniqu {b 1, b 2 } J such that cl MJ {b 1, b 2 }. Th lmnts o J ar usually rrrd to as joints. A undamntal xampl o ram matroids is th ollowing. Givn a ild F, an F-ram matrix is an F-matrix or which vry column has at most two nonzro componnts. Th vctor matroid o a ram matrix is a ram matroid whr th joints ar th columns o an idntity matrix prpndd to th ram matrix. An F-rprsntabl matroid M that may b rprsntd by a ram matrix ovr F w shall rr to as an F-ram matroid. Not vry ram matroid that is F-rprsntabl is an F-ram matroid, howvr. Any ram matroid has a graphical structur calld a biasd graph that was irst xplord by Zaslavsky [13, 14]. Gain graphs (somtims calld voltag graphs, spcially in th ara o topological graph thory) ar a spciic typ o biasd graph that ar spcially usul or studying Dowling gomtris and thir minors. Th rlativ importanc o ram matroids among all matroids was irst displayd by Kahn and Kung [3]. Thy ound that thr ar only two classs o matroid varitis that can contain 3-connctd matroids: simpl matroids rprsntabl ovr GF (q) and Dowling gomtris and thir simpl minors (which ar ram matroids). Mor rcntly th matroidminors projct o Gln, Grards, and Whittl [2] has ound th ollowing ar-raching gnralization o Symour s dcomposition thorm or rgular matroids [8]. I M is a propr minor-closd class o th class o GF (q)-rprsntabl matroids, thn any mmbr o M o suicintly high vrtical connctivity is ithr a boundd-rank prturbation o a ram Dat: Novmbr 20, Mathmatics Subjct Classiication. 05B35. Ky words and phrass. ram matroids, biasd graph, clons, 3-connctd. Slilaty is partially supportd by a grant rom th Simons Foundation #

2 matroid, th dual o a boundd-rank prturbation o a ram matroid, or is rprsntabl ovr som subild o GF (q). Two lmnts and o a matroid M ar said to b clons whn th bijction rom E(M) to E(M) that xchangs and and lavs all othr lmnts ixd is an automorphism o M. Hnc clon rlation on th lmnts o M is an quivalnc rlation whos quivalnc classs ar calld clonal classs. Clons in matroids ar important in studying inquivalnt rprsntations o matroids ovr init ilds [1] and, as mntiond in th prvious paragraph, so ar ram matroids. In this papr w will dtrmin th structur o clonal classs o 3- connctd ram matroids in trms o th structur o biasd graphs (Thorms 2.5 and 2.6). Robbins (th irst author listd in this papr) has conjcturd in [5] that a 3-connctd, nonuniorm matroid M with a clonal class o siz q 1 is not GF (q)-rprsntabl. Robbins conjctur is known to b tru [5, 6] or q {2, 3, 4, 5, 7, 8} and in this papr w will conirm th conjctur or th class o ram matroids (Thorm 2.7). 2. Prliminaris and statmnts o main rsults Givn a graph G, th st o dgs o G is dnotd by E(G). An dg with two distinct ndpoints is calld a link and an dgs with only on ndpoint is calld a loop. Givn X E(G), by V (X) w man th st o vrtics in G that ar incidnt to dgs in X and by G:X w man th subgraph o G with dgs X and vrtics V (X). A k-sparation o G is a bipartition (X, Y ) o th dgs o G or which X k, Y k, and V (X) V (Y ) = k. A connctd graph G is said to b k-connctd whn it has no t-sparation or t < k. A connctd graph G is said to b vrtically k-connctd whn ithr G has at last k + 2 vrtics and thr is no st o t < k vrtics whos rmoval lavs a disconnctd subgraph or G has k + 1 vrtics and contains a spanning complt subgraph. W rquir th notion o an H-bridg in a graph. Lt H b a subgraph o G. Thn an H- bridg is ithr an dg not in H whos ndpoints ar both in H; or a connctd componnt o G\V (H) along with th dgs that link that componnt to H. A Θ-subgraph o a graph G is a 2-connctd subgraph with xactly two vrtics o dgr 3 and all othr vrtics having dgr 2. Rr to Figur 1 or an illustration. A biasd graph is pair (G, B) whr G is a graph and B is a collction o cycls o G such that vry Θ-subgraph o G has 0, 1, or 3 cycls in B. Cycls in B ar calld balancd cycls. Th biasd graph (G, B) is said to b balancd whn vry cycl is balancd; othrwis (G, B) is said to b unbalancd. Whn a biasd graph has no balancd cycls it is said to b contrabalancd. A subgraph o (G, B) is a pair (H, B ) or which H is a subgraph o G and B is B rstrictd to th cycls o H. Somtims w rr to such a subgraph as th subgraph o (G, B) on H. A balancing st o (G, B) is a collction o dgs o (G, B) whos rmoval lavs a balancd subgraph. Th ram matroid M(G, B) is th matroid with ground st E(G) and a circuit is th dg st o ithr a balancd cycl, a contrabalancd handcu or a contrabalancd Θ-subgraph (s Figur 1). Graphic matroids, signd-graphic matroids, and bicircular matroids ar all spcial cass o ram matroids: or graphic matroids, all cycls ar balancd; in a signdgraphic matroid, vry Θ-graph contains ithr 1 or 3 balancd cycls; and in a bicircular matroid, vry Θ-graph contains 0 balancd cycls. Givn a subst X E(G), th rank o X in M(G, B), dnotd r(x), is V (X) b X in which b X is th numbr o connctd componnts o G:X that ar balancd. 2

3 Givn a biasd (G, B), i w add an unbalancd loop at ach vrtx o G and dnot th nw biasd graph by (Ḡ, B). Lt L = {l v v V (Ḡ) and l v is an unbalancd loop at v}. Thn th st L can b rgardd as th st o joints in th ram matroid M(Ḡ, B); as vry link (i.., an dg that is not a loop) is spannd by th two unbalancd loops at its two nd vrtics. (a) (b) (c) Figur 1. A Θ-graph is a subdivision o th graph (a), a tight handcu is a subdivision o th graph (b), and a loos handcu is a subdivision o th graph (c). W will charactriz clonal classs in 3-connctd ram matroids. W us A B to dnot th symmtric dirnc o two sts A and B. Proposition 2.1 ollows immdiatly rom th dinition o clons. Proposition 2.1. Two lmnts and in a matroid M orm a clonal pair i and only i or ach circuit C o M with C {, } = 1 w hav that C {, } is also a circuit. Using th orm o th rank unction or M(G, B) w immdiatly gt Proposition 2.2. Proposition 2.2. I (G, B) is connctd and unbalancd, thn an dg in (G, B) is a coloop o M(G, B) i ithr is an isthmus and th subgraph on G\ consists o on balancd connctd componnt and on unbalancd connctd componnt or is a balancing dg o (G, B). Our analysis o biasd graphs rquirs charactrizing 3-connctivity o ram matroids in trms o th structur o th biasd graphs. Givn a biasd graph (G, B) and a k-sparation (X, Y ) o G, w call th subgraphs o (G, B) on G:X and G:Y, th sids o th sparation. Proposition 2.3 (Qin, Slilaty [10, Thorm 1.4]). I (G, B) is a connctd and unbalancd biasd graph with at last thr vrtics, thn M(G, B) is 3-connctd i (G, B) is vrtically 2-connctd, vry 2-sparation has both sids unbalancd, vry 3-sparation has at last on sid unbalancd, thr ar no balancd cycls o lngth 1 or 2, thr ar no two loops incidnt to th sam vrtx, and thr is no balancing st o 1 or 2 dgs. A vrtx u in a biasd graph (G, B) is calld a balancing vrtx i th subgraph on G\u is balancd. Proposition 2.4 (Zaslavsky [12]). Lt (G, B) b an unbalancd biasd graph whr G is connctd and lt X V (G) b th collction o balancing vrtics o (G, B). I X 3, thn (G, B) has th structur shown on th right in Figur 2 whr th vrtics shown ar th vrtics o X and th shadd rgions ar balancd subgraphs. 3

4 Furthrmor, a cycl o (G, B) is unbalancd i it uss dgs rom mor than on shadd rgion. I X = 2, thn (G, B) has th structur shown on th lt in Figur 2 whr th vrtics shown ar th vrtics o X and th shadd rgions ar balancd subgraphs. Furthrmor, a cycl o (G, B) is unbalancd i it uss dgs rom mor than on shadd rgion. Figur 2. Structur o biasd graphs with mor than on balancing vrtx. Thorm 2.5 is th irst rsult o this papr. Thorm 2.5. Lt (G, B) b a biasd graph such that M(G, B) is 3-connctd. I and ar clons in M(G, B), thn on o th ollowing holds. and ar paralll links in G and vry cycl C o G with C {, } = 1 is unbalancd; and ar loops at two distinct vrtics in G and M(G, B) is isomorphic to U 2,n ; is a loop and is a link in G and ithr M(G, B) is isomorphic to U 3,5, M(G, B) is isomorphic to U 2,n, or (G, B) has th irst structur shown in Figur 3. and ar both links in G and ithr M(G, B) is isomorphic to U 4,6, (G, B) has th scond or third structur shown in Figur 3, or (G, B) has 3 or 4 vrtics and is on o th biasd graphs rom Figur 4. 4

5 v u v u v g x u x Figur 3. Thr structurs or (G, B) supporting a clonal pair, whr, in ach, th shadd rgion rprsnts a balancd subgraph. In th irst structur, thr may b links paralll to and vry cycl containing is unbalancd. In th scond structur, th dgr o v is xactly thr, vry cycl containing but not containing is unbalancd, and (G\v, B ) is unbalancd; morovr, i thr is no balancd cycl containing both and, thn,, and g will orm a clon tripl. In th third structur, any cycl not containd compltly within th shadd rgion is unbalancd. Figur 4. Som small structurs whr vry loop and vry digon is unbalancd; and in th third structur, vry triangl containing but not containing is unbalancd. As corollaris to Thorm 2.5 w hav Thorms 2.6 and 2.7. Th lattr is conjcturd by Robbins in [5] to hold or all 3-connctd matroids and not just th class o 3-connctd ram matroids. A clonal class X in M(G, B) is said to b a canonical clonal class whn ithr X is a collction o paralll dgs in (G, B) or which no dg is in a balancd cycl or X\ is such a collction o paralll dgs with bing an unbalancd loop incidnt to th dgs in X\. Th nxt thorm ollows immdiatly rom Thorm 2.5. Thorm 2.6. Lt (G, B) b a biasd graph with at last thr vrtics such that M(G, B) is 3-connctd and M(G, B) = U3,5, U 3,6, or U 4,6 and lt X b a clonal class o lmnts in M(G, B) with X 3. On o th ollowing holds. X is a canonical clonal class. X = 4, G has our vrtics, and (G, B) has th structur shown in Figur 5. X = 3 and (G, B) has on o th structurs shown in Figur 6. 5

6 g h Figur 5. A ram matroid with a clonal class o siz 4, whr {g, h} is a paralll class o siz xactly two and th graph is contrabalancd. v g u u x g Figur 6. Thr structurs or (G, B) supporting a clonal tripl, whr, ach shadd rgion rprsnts a balancd subgraph. In th irst structur, any cycl not containd compltly within th shadd rgion is unbalancd. In th scond structur, th biasd graph is contrabalancd. In th third structur, th dgr o v is xactly thr, (G\v, B ) is unbalancd, and thr is no balancd cycl containing th dg. v g x Thorm 2.7. I M(G, B) is 3-connctd, not a uniorm matroid, and contains a clonal class o siz q 1 2, thn M(G, B) is not GF (q)-rprsntabl. Proo. Th rsult is known or q {3, 4, 5} or gnral matroids and not just ram matroids [5]. So assuming that q 7, w gt that any clonal class in M(G, B) o siz at last q 1 is a canonical clonal class by Thorm 2.6. I X is a canonical clonal class in M(G, B) is siz at last q 1, thn th act that M(G, B) is 3-connctd and non-uniorm orcs G to hav at last thr vrtics. On can chck that (G, B) now contains a minor rom Figur 7 in which ach biasd graph shown is contrabalancd. Figur 7. Th clonal class X is th collction o dgs on th bottom two vrtics o ach coniguration. Ths thr biasd graphs hav ram matroids that ar obtaind by applying a Y opration to U 2,l or l q + 2 and this matroid is not GF (q)-rprsntabl. 6

7 3. Proo o Thorm 2.5 First suppos and ar paralll links. I {, } is not a balancd cycl, thn and ar clons i vry cycl C with C {, } = 1 is unbalancd. For th rst o th sction, w will assum that and ar not paralll links. Suppos that and ar both loops not at th sam vrtx and nithr is balancd. Assum is incidnt with u and is incidnt with v. Now i and ar clons, thn both G\u\ and G\v\ ar balancd. Thror, both u and v ar balancing vrtics o G\{, }. So (G\{, }, B) has on o th structurs o Figur 2. Sinc M(G, B) is 3-connctd, ach shadd rgion must b a singl link, and hnc, M(G, B) must b isomorphic to U 2,n. Nxt assum that is a loop and is a link. I and ar clons, must b unbalancd and, urthrmor, no balancd cycl o (G, B) contains. Suppos that is incidnt to u and is incidnt to v and w. W may assum that u v. Sinc M(G, B) is 3-connctd, G\v is a connctd graph, G\v must b balancd atr rmoving th loop as othrwis thr would b a handcu in G\v containing th loop, contrary to th act that and ar clons. I G\v has xactly on vrtx, thn M(G, B) is isomorphic to U 2,n. Assum that V (G\v) 2. Considr th ollowing two cass. Cas 1: u w. Not that G\ is vrtically 2-connctd. I G\ has an unbalancd cycl not using v or w, thn by th 2-connctivity and th act that vry cycl containing is unbalancd, w can construct a Θ-subgraph containing in G\, say Θ 1. Sinc u / {v, w} by th assumption, Θ 1 \{} {} is not a circuit in M(G, B), contrary to th act that and ar clons. Thror, both v and w ar balancing vrtics o G\. So G\ has on o th structurs o Figur 2. Morovr, it ollows rom th 3-connctivity o M(G, B) that ach {v, w}-bridg not containing th vrtx u must b a singl link. Thr ar at most two such links including ; as othrwis, w can ind a Θ-graph, say Θ 2, containing and two othr links paralll to. Clarly Θ 2 \{} {} is not a circuit o M(G, B), a contradiction. Sinc M(G, B) is 3-connctd, {, } is not a sris pair, and hnc, G\{, } is unbalancd. So thr must b xactly two links btwn v and w including. Now in th {v, w}-bridg that contains u, u must b a cut vrtx. It ollows rom th 3-connctivity that G has xactly thr vrtics u, v and w and G consists o a pair o paralll links btwn v and w, a loop at u, a link btwn u and v, and a link btwn u and w. Sinc M(G, B) is 3-connctd, (G, B) contains no balancd cycls, and hnc, M(G, B) is isomorphic to U 3,5. Cas 2: u = w. Now G\v\ must b balancd and is not in any balancd cycl and G has th irst structur shown in Figur 3. Finally assum both and ar links. Suppos that is incidnt with u and v, and is incidnt with w and x. Sinc and ar not paralll to ach othr, w may assum that u / {w, x}. Sinc M(G, B) is 3-connctd, G is vrtically 2-connctd, and hnc, G\u is connctd. Sinc and ar clons, is a coloop in M(G\u, B ). Thn ithr is an isthmus or is a balancing dg o (G\u, B ). (S Proposition 2.2.) Cas 1: is an isthmus o G\u. 7

8 Lt G 1 and G 2 b th two componnts o G\u\. Assum that w G 1 and x G 2. Furthrmor and without loss o gnrality w may assum that G 1 is balancd. Considr th ollowing two subcass: Subcas 1.1: G 1 has mor than on vrtx. In this cas irst assum v G 1. Sinc thr is a 2-sparation o G at {u, w}, th 3- connctivity o M(G, B) implis that th {u, w}-bridg G 1 that contains G 1 is unbalancd. Hnc u is a balancing vrtx o G 1 and so thr is an unbalancd cycl C in G 1 that contains. Similarly th {u, x}-bridg G 2 that contains G 2 is unbalancd and thr is an unbalancd cycl C 2 in G 2. Thror thr xists a handcu H in G that contains C C 2 and dos not us th dg. It ollows that v = w sinc v has dgr-1 in H \{} and has dgr at last 2 in H {, }. Now sinc M(G, B) is 3-connctd, thr xists an unbalancd cycl C u in G 1 not containing. Hnc w can ind a Θ-subgraph Θ 1 containing and C u. Not that sinc and ar clons and thy ar not paralll, vry cycl containing and not containing is unbalancd. So Θ 1 is a circuit o M(G, B). Now clarly Θ 1 {, } is not a circuit o M(G, B), contrary to th act that and ar clons. Thror w must hav that v G 2. Again in th {u, x}-bridg G 2 that contains G 2, thr xists an unbalancd cycl C 1 ; and in th {u, w}-bridg G 1 that contains G 1, thr xists an unbalancd cycl C u containing th vrtx u. Clarly thr xists a handcu H containing C 1 C u {}. Sinc and ar clons, th cycl C u must us th vrtx w. Thror, both u and w ar balancing vrtics or G 1. By Proposition 2.4, G 1 has on o th structurs shown in Figur 2. So sinc G 1 has at last thr vrtics, (G, B) has a 2-sparation with on sid balancd, contradicting th 3-connctivity o M(G, B). Subcas 1.2: G 1 has on vrtx. In this cas, w is th only vrtx o G 1. I v = w, thn by th 3-connctivity o M(G, B) and th act that and ar clons, thr xists a uniqu link g paralll to and th digon {, g} is unbalancd. Sinc and ar clons, vry cycl containing and g must b unbalancd. Furthrmor, in th {u, x}-bridg G 2 that contains G 2, x must b a balancing vrtx: or othrwis thr xists a unbalancd cycl C 4 in G 2 not using x and w can construct a handcu H using C 4 and th digon {, g}, and not using x. Clarly H {, } is not a circuit o M(G, B), a contradiction. Thror (G, B) has th scond structur shown in Figur 3. I v G 2, thn thr xists a pair o links g and h btwn u and w and th digon {g, h} must b unbalancd. Now considr th {u, x}-bridg G 2 that contains G 2. First assum thr xists an unbalancd cycl C not using v. I u / V (C), thn w can ind a handcu H containing C and {, g, h} and not using v. Clarly H {, } is not a circuit o M(G, B), a contradiction. I u V (C), thn w can ind a Θ-graph Θ containing C and th link. Again Θ {, } is not a circuit o M(G, B); a contradiction. Thror v is a balancing vrtx o G 2. Similarly w can show x is a balancing vrtx o G 2. Again by 3-connctivity and Proposition 2.4 that G 2 has ithr two or thr vrtics. Thror G has thr or our vrtics. A routin cas analysis shows G must b on o graphs shown in Figur 4. Not that in th third graph, vry cycl containing but not containing must b unbalancd. Cas 2: is a balancing dg o (G\u, B ). 8

9 Not that w may urthr assum that is a balancing dg o (G\w, B ). Thror, vry unbalancd cycl o G ithr uss th dg or uss th vrtx u; similarly, vry unbalancd cycl o G ithr uss th dg or th vrtx w. So in th graph G\{, }, both u and w ar balancing vrtics. So G\{, } has on o th structurs rom Figur 2. I and li in two dirnt {u, w}-bridgs, thn G\{, } must hav xactly two {u, w}- bridgs; on contains and th othr contains. (As othrwis, w can ind a contrabalancd Θ-subgraph containing only; and rmoving and adding dos not produc a Θ-subgraph or a handcu, contrary to th act and ar clons.) Similarly w can ind a contrabalancd Θ-subgraph C containing such that C {, } is not a circuit i x is not a cut vrtx o th {u, w}-bridg containing. Thus x is a cut vrtx o {u, w}-bridg containing and also v is a cut vrtx o {u, w}-bridg containing. It ollows rom th 3-connctivity o M(B, G) that G has xactly our vrtics and six dgs and M(G, B) is isomorphic to U 4,6. I and li in th sam {u, w}-bridg, say G 1, thn by th 3-connctivity, all othr {u, w}-bridgs contain a singl link. Not that i v = x, thn (G, B) has th third structur shown in Figur 3. So now assum that v x. By Cas 1 and symmtry, w may assum that is a balancing dg o G\x and is a balancing dg o G\v. It ollows that G\{, } has xactly two {u, w}-bridgs: on that contains th vrtics v and x, and th othr consists o a singl link, say g, conncting u and w. Morovr, {,, g} is a balancing st or (G, B). Sinc M(G, B) is 3-connctd, G\{, } is not balancd, and hnc contains an unbalancd cycl C g. Clarly C g contains th link g and a path rom u to w. I v is not on th cycl C g, thn w can ind a path rom v to th cycl C g, thus construct a contrabalancd Θ-subgraph Θ containing. Sinc v x, v has dgr 1 in Θ {, }, contrary to th act that and ar clons. By symmtry, C g must also contain th vrtx x. Thror, in G\g, vry path rom u to w must contain both v and x. Considr th {u, w}-path C g \g and lt u b th starting vrtx. Thn ithr th vrtx v coms bor x or it coms atr th vrtx x. In th ormr cas, sinc M(G, B) is 3- connctd, th path C g \g consists o xactly thr links, rom u to v, rom v to x, and rom x to w, and G has xactly our vrtics and six dgs, and G is th graph obtaind rom a pair o disjoint digons connctd by a matching. It is routin to chck that M(G, B) is isomorphic to U 4,6. In th lattr cas, it ollows rom th 3-connctivity o M(G, B) again that G is isomorphic to K 4 and M(G, B) is isomorphic to U 4,6. Rrncs [1] Gln J. F., Oxly, J.G., Vrtigan, D., and Whittl,G. P., Totally r xpansions o matroids, J. Combin. Thory Sr. B 84 (2002), no. 1, [2] Gln, J. and Grards, B. and Whittl, G., Solving Rota s conjctur, Notics Amr. Math. Soc. 61 (2014) no. 7, [3] Kahn, J. and Kung, J. P. S. Varitis o combinatorial gomtris, Trans. Amr. Math. Soc. 271 (1982) no. 2, [4] Oxly, J.G., Matroid Thory, Oxord Univ. Prss, Nw York (1992). [5] Rid, T.J., Robbins, J., A not on clon sts in rprsntabl matroids, Discrt Math. 306 (2006), [6] Rid, T.J., Zhou, X., On clon sts o GF (q)-rprsntbal matroids, Discrt Math 309 (2009), [7] Rid, T.J.; Robbins, J.; Wu, H.; Zhou, X., Clonal sts in GF(q)-rprsntabl matroids, Discrt Math. 310 (2010), [8] Symour, P.D., Dcomposition o Rgular Matroids, J. Combin. Thory Sr. B 28 (1980),

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