ON CUBIC AND EDGE-CRITICAL ISOMETRIC SUBGRAPHS OF HYPERCUBES

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1 Univeity of Ljbljn Intitte of Mthemtic, Phyic nd Mechnic Deptment of Mthemtic Jdnk 19, 1000 Ljbljn, Sloveni Pepint eie, Vol. 40 (2002), 824 ON CUBIC AND EDGE-CRITICAL ISOMETRIC SUBGRAPHS OF HYPERCUBES C. Pl Bonnington Sndi Klvž Alenk Lipovec ISSN My 6, 2002 Ljbljn, My 6, 2002

2 On cbic nd edge-citicl iometic bgph of hypecbe C. Pl Bonnington Deptment of Mthemtic Univeity of Acklnd Acklnd, New Zelnd p.bonningtoncklnd.c.nz Sndi Klvz y Deptment of Mthemtic, Univeity of Mibo Kook cet 160, 2000 Mibo, Sloveni ndi.klvzni-lj.i Alenk Lipovec Deptment of Edction, Univeity of Mibo Kook cet 160, 2000 Mibo, Sloveni lenk.lipovecni-mb.i Abtct All cbic ptil cbe (i.e., cbic iometic bgph of hypecbe) p to 30 vetice nd ll edge-citicl ptil cbe p to 14 vetice e peented. The lit of gph wee conmed by compte ech to be complete. Non-tivil cbic ptil cbe on 36, 42, nd 48 vetice e lo contcted. 1 Intodction Ptil cbe e, by denition, gph tht dmit iometic embedding into hypecbe. They wee intodced by Ghm nd Pollk [10] nd t chcteized by Djokovic [7]. Sevel dditionl chcteiztion followed in [2, 5, 19, 20]. Ptil cbe fond dieent ppliction (ee, fo exmple, [6, 8, 13]), while ecognition lgoithm fo thee gph hve been developed in [1, 11]. Fo n extenive peenttion of ptil cbe we efe the ede to the book [12]. Sppoted by the Mden Fnd (Gnt Nmbe UOA-825) dminiteed by the Royl Society of New Zelnd. y Sppoted by the Minity of Edction, Science nd Spot of Sloveni nde the gnt

3 Fo the (pobbly) mot impotnt bcl of ptil cbe, medin gph, Mlde [17] poved tht hypecbe e the only egl medin gph. In othe wod, the only egl medin gph e Ctein podct of copie of K 2. Thi elt h been in [4] extended to the o-clled \tee-like" ptil cbe. Hence, it i ntl to k which gph e egl ptil cbe. (Regl bgph of hypecbe e tdied in [18]). Depite the fct tht the tcte of ptil cbe h been well clied by now, thi qetion eem to be diclt one. The Ctein podct of two (egl) ptil cbe i (egl) ptil cbe. Since even cycle e egl ptil cbe, one my wonde whethe we get ll egl ptil cbe Ctein podct of copie of K 2 nd even cycle. In pticl, e ll cbic ptil cbe of the fom C 2k 2K 2, k 2? Thi w believed to be te fo qite while, ntil two podic exmple ppeed: the genelized Peteen gph P (10; 3) on 20 vetice, cf. [14], nd the gph B 1 (ee Fig. 1) on 24 vetice fom [9], (ee lo [12]). Clling the gph C 2k 2K 2, k 2, tivil cbic ptil cbe, we hve veied tht beide thee two gph thee i only one othe nontivil cbic ptil cbe on t mot 30 vetice. The thid exmple, denoted B1 0 (ee Fig. 2), h 30 vetice. It cn be obtined fom the nontivil ptil cbe on 24 vetice by the o-clled expnion nd w lo fond by compte ech. B 1 B 2 Fige 1: Gph B 1 nd B 2 Edge-citicl ptil cbe e ptil cbe G fo which G e i not ptil cbe fo ll edge e of G. The 3-cbe nd the bdiviion gph of K 4 e the only edge-citicl ptil cbe on t mot 10 vetice [15]. In thi note we peent ll cbic ptil cbe p to 30 vetice nd ll edge-citicl ptil cbe p to 14 vetice. The lit of gph wee conmed by compte ech 2

4 to be complete. We lo give fthe lge non-tivil cbic ptil cbe on 36, 42, nd 48 vetice. 2 Cbic ptil cbe A gph G i clled pime (with epect to the Ctein gph podct) if G = G 1 2G 2 implie eithe G 1 o G 2 i the one-vetex gph K 1. The Ctein podct of two egl ptil cbe i egl ptil cbe. Theefoe the poblem of chcteizing egl ptil cbe edce to pime (with epect to the Ctein podct) ptil cbe. Fo the cbic ce, thi fct led to the following obevtion: Popoition 2.1 Let G be cbic ptil cbe. Then eithe G = C 2n 2K 2 fo ome n 2 o G i pime gph. Poof. Ame G = G 1 2G 2, whee G 1 ; G 2 6= K 1. A G i connected, then o e G 1 nd G 2. Since G i cbic nd the degee of (; v) 2 V (G 1 2G 2 ) i the m of the degee of 2 G 1 nd v 2 G 2, then one of the fcto, y G 2, contin only vetice of degee one o le. Theefoe G 2 = K 2. Fthemoe, G 1 mt be 2-egl, nd hence cycle. Moeove, it i n even cycle ince ptil cbe e biptite gph. 2 We now contct the nontivil cbic ptil cbe B1, 0 B1 00, B1 000, nd B 2 on 30, 36, 42, nd 48 vetice, epectively. The lte gph i hown in Fig. 1, while the othe e given in in Fig. 2. Thee gph cn be contcted by expnion fom B 1, nd hence we t intodce the concept of expnion. Let G 0 be connected gph. A pope cove conit of two iometic bgph G 0 1 ; of G0 2 ch tht G G0 0 = G 0 1 [, = G0 2 G0 0 G0 1 \ G0 2 i nonempty bgph, nd thee e no edge between G 0 1 n G0 2 nd G 0 2 n 1. (The bgph G G0 0 0 i clled the inteection of the cove.) The expnion of G 0 with epect to G 0 1, G 0 2 i the gph G contcted follow: Let G i be n iomophic copy of G 0 i, fo i = 1; 2, nd, fo ny vetex 0 in G 0 0, let i be the coeponding vetex in G i, fo i = 1; 2. Then G i obtined fom the dijoint nion G 1 [ G 2, whee fo ech 0 in G 0 0 the vetice 1 nd 2 e joined by n edge. Chepoi [5] poved tht gph i ptil cbe if nd only if it cn be obtined fom K 1 by eqence of expnion. Thi elt w lte independently obtined in [8] nd i nlogo to the convex expnion theoem fo medin gph [16]. An expnion i clled peiphel if t let one of the gph G 0 1 o G 0 2 i eql to G. In thi ittion the othe gph eql the inteection, nd i neceily iometic 3

5 f f f f f f B 1 f f f f B 00 1 f f f f f f f B 0 1 B f f f f f f f Fige 2: Gph B 1, B 0 1, B 00 1, nd B in G. We ecll fom [4] tht egl, pime ptil cbe on t let thee vetice cn not be obtined by peiphel expnion fom ome ptil cbe. Fo the poof of the next elt we lo need the following concept of iometic dimenion. Two edge e = xy nd f = v of gph G e in the Djokovic-Winkle [7, 20] eltion if d G (x; )+d G (y; v) 6= d G (x; v)+d G (y; ). Winkle [20] howed tht biptite gph i ptil cbe if nd only if = (whee denote the tnitive cloe of ). Th dene n eqivlence eltion on the edge of ptil cbe. The iometic dimenion, idim(g), of ptil cbe G i dened the nmbe of it -cle. Theoem 2.2 Gph B 0 1, B 00 1, B 000 1, nd B 2 e cbic pime ptil cbe. 4

6 Poof. We know ledy tht B 1 i ptil cbe. Now, B1 0, B 1 00, B 1 000, nd B 2 cn be obtined fom B 1, B1, 0 B1 00, nd B1 000, epectively, by n expnion. Thee expnion e chemticlly explined in Fig. 2 in the following wy. A pope cove in ech expnion i choen follow: G 0 1 i indced by the vetice denoted by lled cicle, G 0 2 i indced by the vetice denoted by lled qe nd thei inteection i fomed by the emining vetice; tht i, the vetice denoted by lled cicle onded by nothe cicle. It i ey to veify tht in thi wy we elly obtin pope cove; tht i, G 0 1 nd G 0 2 e iometic bgph of the coeponding gph B 1, B1, 0 B1 00, nd B1 000, nd thee e no edge between G 0 1 n G0 2 nd G 0 2 n 1. Hence, by the theoem of G0 Chepoi the obtined gph e ptil cbe. Clely, they e cbic. We now how tht thee fo gph e pime. Obeve t tht idim(b 1 ) = 6 nd theefoe idim(b1) 0 = 7, idim(b1 00 ) = 8, idim(b1 000 ) = 9, nd idim(b 2 ) = 10. If ny of thee fo gph wee not pime, then by Popoition 2.1 it wold be iomophic to C 15 2K 2, C 18 2K 2, C 21 2K 2, nd C 24 2K 2, epectively. Two of thee gph e not biptite, while the iometic dimenion of the othe two; tht i, of C 18 2K 2, nd C 24 2K 2, e 10 nd It eem tempting to contine the expnion pocede with B 2 to obtin new cbic ptil cbe. Howeve, we wee not ble to obtin moe exmple in thi wy. In pticl, the gph tht i contcted fom B 2 nlogoly B 2 i contcted fom B 1 i not ptil cbe. 3 Compte ech fo cbic nd edge-citicl ptil cbe Uing the Djokovic-Winkleeltion, we hve implemented ecognition lgoithm fo ptil cbe nd pplied it to ll connected biptite cbic gph p to 30 vetice. (Thee gph wee contcted ing Bendn McKy' Nty pogm [3].) The exmintion of the entie et of gph w n concently on clte of 16 pentimcl mchine, nd dobled-checked on 8 poceo Sn Spc eve. The obtined elt e mmized in the following tble: n n < 8-20 C 10 2K 2 P (10; 3) 8 C 4 2K C 12 2K 2 B 1 12 C 6 2K C 14 2K 2-16 C 8 2K B

7 The bove tble how tht, p to 30 vetice, thee e only 3 nontivil cbic ptil cbe. Recll tht ptil cbe G i clled edge-citicl if fo ny edge e of G, G e i not ptil cbe. In [15] it w hown tht the 3-cbe Q 3 nd the bdiviion gph of K 4, S(K 4 ), e the only edge-citicl ptil cbe on t mot 10 vetice. Moeove, two ch gph on 12 vetice nd one on 13 vetice e lited. We hve now eched fo ll edge-citicl ptil cbe on t mot 14 vetice nd etblihed the following complete lit of edge-citicl ptil cbe. The compttion i vint of tht ed fo cbic ptil cbe; fo ech connected biptite gph G on t mot 14 vetice, if G i detemined to be ptil cbe, then ll of the non-iomophic gph obtined by deleting ingle edge fom G e teted. Bendn McKy' Nty pogm [3] i ed to lo lte iomophic gph fom the edge deletion. The elt e mmized in the following tble. n < { Q 3 { S(K 4 ) E 1 E 2, E 3, E 4 E 5 E 7 ; : : : ; E 12 We note tht thee i one pevioly ndicoveed gph on ech of 11 nd 12 vetice (E 1 nd E 2 epectively), nd ix on 14 vetice. Thee e given in Fig. 3 nd 4. E 1 t t t t t t t t t t t E4 H H H H E 2 E 5 E 3 QQ Fige 3: Edge-citicl ptil cbe on 11, 12, nd 13 vetice 6

8 ### #,,,, % e e % e e % % e % e % l, l, l, l, l l,, l l QQ Q Q " "" H b bb " ""H!!! l S ll S l S ll " "" H HH Q Q!!! S SS J JJ S J SS e e % l, l,, hhhhhhh h,, ((((((((((( % % % %, Fige 4: Edge-citicl ptil cbe on 14 vetice Refeence [1] F. Aenhmme nd J. Hge, Recognizing biny Hmming gph in O(n 2 log n) time, Mth. Sytem Theoy 28 (1995) 387{395. [2] D. Avi, Hypemetic pce nd the Hmming cone, Cnd. J. Mth. 33 (1981) 795{802. [3] B. McKy, NAUTY e' gide, Technicl epot TR-CS-90-02, Comp. Sci. Dept., Atlin Ntionl Univeity. [4] B. Be, W. Imich, S. Klvz, Tee-like iometic bgph of hypecbe, bmitted. 7

9 [5] V. Chepoi, d-convexity nd iometic bgph of Hmming gph, Cybenetic 1 (1988) 6{9. [6] V. Chepoi nd S. Klvz, The Wiene index nd the Szeged index of benzenoid ytem in line time, J. Chem. Infom. Compt. Sci. 37 (1997) 752{755. [7] D. Djokovic, Ditnce peeving bgph of hypecbe, J. Combin. Theoy Se. B 14 (1973) 263{267. [8] K. Fkd nd K. Hnd, Antipodl gph nd oiented mtoid, Dicete Mth. 111 (1993) 245{256. [9] E. Gedeonov, Contction of S-lttice, Ode 7 (1990) 249{266. [10] R.L. Ghm nd H. Pollk, On the ddeing poblem fo loop witching, Bell Sytem Tech. J. 50 (1971) 2495{2519. [11] W. Imich, nd S. Klvz, A imple O(mn) lgoithm foecognizing Hmming gph, Bll. Int. Comb. Appl. 9 (1993) 45{56. [12] W. Imich nd S. Klvz, Podct Gph: Stcte nd Recognition, (Wiley, New Yok, 2000). [13] S. Klvz nd I. Gtmn, Wiene nmbe of vetex-weighted gph nd chemicl ppliction, Dicete Appl. Mth. 80 (1997) 73{81. [14] S. Klvz nd A. Lipovec, Ptil cbe bdiviion gph nd genelized Peteen gph, Pepint Se. Dept. Mth. Univ. Ljbljn, vol. 39 (2001) no Alo: Dicete Mth., to ppe. [15] S. Klvz nd A. Lipovec, Edge-citicl iometic bgph of hypecbe, Pepint Se. Dept. Mth. Univ. Ljbljn, vol. 39 (2001) no Alo: A Combin., to ppe. [16] H.M. Mlde, The tcte of medin gph, Dicete Mth. 24 (1978) 197{204. [17] H. M. Mlde, n-cbe nd medin gph, J. Gph Theoy 4 (1980) 107{110. [18] M. Rm, Regl bgph of hypecbe, A Combin. 52 (1999) 21{32. [19] E. Wilkeit, Iometic embedding in Hmming gph, J. Combin. Theoy Se. B 50 (1990) 179{197. [20] P. Winkle, Iometic embedding in podct of complete gph, Dicete Appl. Mth. 7 (1984) 221{225. 8

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