MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH Tag u n 9 s b e r i c ht ")1/1984 Graphentheorie 8.7. bis 14.7.1984 Die Tagung fand unter der Leitung von' Herrn' G. Ringel (Santa Cruz, California) und Herrn W. Mader (Hannover) statt. Mehr als 40 Teilnehmer aus 13 Ländern (Australien, 'VR " China, CSSR, Dänemark, England, Frankreich, ISI;'ael,.·· Jugoslawien, Kanada, Niederlande, Ungarn, USA) haben über ihre n~uesten Ergebnisse in verschiedenen Gebieten der Graphentheorie b~richtet. In einer von Herrn P. Er~' dös geleiteten Problemsitzung wurden offene Probleme' vorgestellt und diskutiert. Im besonderen wurden Resultate aus den folgenden Teil- ~ gebieten der Graphentheorie vorgetragen: Färbungsprobleme, (Kanten- bzw. Knotenfärbunge~/·Larid~· ka.rten) , zusammennangsprobleme, Unendliche. Graphen, Einbettungen.von Graphen in verschiedene Fiächen, (Mini­ malbasen, Kantenkreuzungen), Partitionen (Turan-Graphen, Ramsey-Theorie usw.), . ~. .. '\ Wege und Kreise in Graphen. Darüber hinaus wurden graphentheoretische Methoden be- © nutzt im Zusammenhang mit Matroiden, Netzwerken, Opti­ mierungsproblemen,' physikalischen Problemen, Lateini­ schen Rechtecken. Als spezielle Ergebnisse seien noch die Fortschritte im "Oberwolfacher Problem" erwähnt sowie die Lösung der seit 1890 offenen Imperium-Vermutung von Heawood. Die Grundidee zu dieser Lösung ist in der abgebildeten Land­ karte mit 19 paarweise benachbarten, aus jeweils 4 ge­ trennten Ländern bestehenden Staaten angedeutet. © - 3 - vortragsauszüge- R. AHARONI: Matchings in infinite graphs A criterion is present~d for a graph· at" any cardinality to passess a perfeet matc~ing. B. ALSPACH: Same new results on. the Oberwolfach problem Let F(11,12, •.• ,lr) IG denate an isomorphi~ factorization of G- into 2-factors each of which is compased of cycles af lengths l1,12, ••• ,lr. If a~l the li'5 are the same value 1, write F(l»)G instead. The results mentioned in this talk are'the following. Thm.1. F(2m) IK2~-I for all positive -integers r~1 and ~2 where I 'de'notes a·. 1-factor of K2~. Thm.2 •. If ~i~4 and even for each i, r th~n F(11.,12' .•• 'lr)IK4m,4~ for every m'~ 1 with L 1. Sm. i=1 1 Thm.·3. odd and n odd, then F .(d~_) 1K If _F(s) IKn , ~ dn for all odd d. ~(s) aso long as s i5 odd, s=9.~m~d3) and Cor.4. IKn n is an pdd multiple of s. The above is joint work .with Roland.Häggkvist. T. ANDREAE : AGame of Cops and Robbers' There are two players c = cop player and r = robber player. First c places s cops at same of the vertices of a finite connected undirected graph, G. (Twa o~ mar~ c~~~ may be on the same vertex. ) Then' r "places ä. robber at some ·vertex. Thereafter the players move alternatively·.·· A move of' c corlsists of moving"some'of the cops along' edges to:adjacent vertice5. Similarly; a move of·r i5 defined. c' 'wins if he catches the robber, r wins if he'· avoids this forever". Let· c (Gl be the minimal number of cops that are sufficient to catch the robber~ AIGNER and FROMME proved that c(G)~3 if G' is planar, and that, © - 4 - in general, c(G) can be arbitrarily high. Here it is shown that, for each finite graph H, there is a minimal a(H)EN such that c(G)~a(H) if H is not a subcontrac­ a(Kn)~(n-1) n~4, tion of G. Further, (n-3) for O(KS) 3, O(R ,3) 3, a(K 2, a(Wn)~rn/31+1 (W wheel = 3 = S) = n = with n rim vertices, Kn(K~) = complete graph with n vertices (minus an e~ge». Other results: 1. QUILLIOT showed c(G)~2n+3 if G has genus n,~. an algorithmic characterization of the graphs with c(G) = 1 is due to QUILLlOT and NOWAKOWSKI/WINKLER, 3. c(G)~2 if !V(G) 1~10, G # Petersen Graph. I. BEN-ARROYO HARTMAN: Path Partitions and Packs in Digraphs A path partititon P {P , ••• ,P } of a digraph is a = 1 m partition of its vertex set inta disjoint directed paths. A path partiti~n P is k-optimal if I: mln{ IP',I ,k} is as P.EP· l. l. small as possible; IPil is the number of vertices in k Pi. A partial k-colouring C is a collection of k dis-, joint independent sets. Berge conjectured the following: Let G be a directed graph and let k be a positive integer. Then for every k-optimal path partition. P there k exists a partial k-colouring C such that each path Pi in P meets min{IPil,k} different colour classes k of c • We report on recent progress on the problem and on a dual analogue, interchanging the roles of paths and colour classes. R. BODENDIEK: Uber ebene Graphen Ein endlicher, ungerichteter, schlichter und in die Kugel Fo einbettbarer Graph" G = (E,K) ist dadurch ausgezeichnet, daß er höchstens einmal gesä~tigt werden kann und daß der Grenzgraph G(L) für jedes Land L der ebenen Landkarte dag~gen (G,Fo-G) ein Husimibaum ist. Ist F * Fo eine orientierbare oder nichtorientierbare Fläche, so lassen sich stets 'in F e1nbettbare Graphen G finden, die min­ destens zweimal gesättigt werden können oder bei denen © - 5 - Länder in der Landkarte (G,F-G) existieren, deren Grenz­ graphen keine Husimibäume·sind. Betrachtet man dagegen die Spindelfläehen o(Cn ), bei.denen der äußere Graph G ein (n~3) jeder'der'i~eren Kreis Cn ist und Graphen -G" 'G2 , ~o Spindel~ ..• , Go ein ,K 2 , kann man zeigen, daß diese flächen für sogenannte normale Graphen, die sich überdies in o(Cn ) so einbetten lassen, daß die' n singulären Punkte von ~(~n) Ecken des eingebetteten Graphen'sind; die gleiche Sättigungs- und Landkarteneigenschaft wie Fa j~doch besitzen. Im Gegensatz zu X(Fo ) = 4 ist die chro-' matische Zahl X(O(~n» für jedes n.?: 3 gleich 5. A. BONDY: Parity Theorems for Paths and Cycles , , , This is areport of joint work with F.Y.' Halberstam. A gra~h is ~ (resp. odd) if every vertex has even (r~s~. od~) d~­ gree. Using a technique of A.G. Thomason, we prove the fol-' 'lowing results. Theorem 1. Le,t , Pi (u) ,be the number of paths }?f length .i with initial vertex u. Then (a~ if G is even, .~~(~) is even (i~1), (~) if G 1s odd, Pi(u) is even~(i,?:2). Theorem ,2. Let Pi be the n~er of paths of leng~~ i . - Then (a) if G is even and bipartite, Pi. is eve~ (i~1), (h) if G 1s odd and bipartite, Pi is even (i~3). Theorem 3. Let v be the.number 'of vertices' and p the number of paths. Then (a) if G 1s even, p = v(mod2), , 1 (b) if G 1s odd, P = 2v(mod2) • Theorem 1 has ari application to the question of exlstence" of generaliz~d friendship graphs (th?se graphs with the' '" • property that, for a given k,?:3, there is a~unique (u,v)-pa,th of length k for all pairs u,v of d1s.tinct vertices)':' A. Kotzig has conjectured that no such graphexists. Theorem 1 (a) imp~1es that each such graph i~ ?f o~~ order. A. BaUCHET: Isotropie systems. Recognizing circle grap~s Let V be a finit~ set. We consider a, 2-dim.ension~l·.yec::tor space K over GF(2) with the non-null alternate bilinear " .. ", . '. 'V'; form (x,y) ....xy and the symplectic structure induced on K © -.6 - by the alternate bilinear form (a,ß)~rVEVa(v)ß(v). An iso­ tropie system tL,V) is defined by a totally isotropie sub­ V spaee cL ~ K of .dimension lVI. Isotropie systems ean be assoeiated to 4-regular graphs ­ ealled graphie systems - and also to pairs of dual binary matroids. In.ge?eral isotropie systems unify some proper­ ties of 4-regular graphs and binary matroids. The application which we develop here comes from the facts that eaeh isotropie system ean be identified to a simple graph defined up to Ioeal eompiementations and t~at ~he cireie graphs (defined by assQeiating vertiees to chords of • a cireie and joining vertices iff the corresponding chords intersect) correspond so to graphie systems. A theory of 3-connectivity for isotropie systems similar to Tutte's theory for matroids allows to derive an efficient algorithm for recognizing circle graphs. P. A. CATLIN: Homomorphisms into add eycles A harnomorphis~ 8: G ~ H is a function fram a graph - G into a graph H such that x - y irnplies Sex) ~ 0(y) , where denotes adjaceney. We eonsider hornomorphism~ as ä generalization of the coneept of vertex coloririg (the case with H eomplete). There is a construetive characterization of the edge-minimal series-parallel graphs having no homomorphism inta eS . For the ease H = C + ' there are variaus. recent results, 2k 1 including a sufficient condition for a homomorphism onto an odd cycle to ~e unique. • G. CHEN: Spanning balanced bipartite subgraphs of regular graphs - On R. HÄGGKVIST's Conjecture This paper points out a counter-example' of R. Häggkvist's conjecture: "Let .G be an rn-regular graph with an even number of vertices, then G contains a spanning balaneed bipartite graph- B where each vertex has degree ~[~]." After modifying the original conjecture, we have proved. that for m ~ 3 the modified conjeeture is true. © - 7 - D. CVETKOVIC: Inte'ractive prograrnming system "Graph" ­ an expert system for graph 'Lheory Interactive programming system "Graph" has recently been implement~d at Faculty of Electrical Engine~ring, University of Belgrade. System "Graph" contain~ a c~mputarizeq.grap~ theory biblio~raphy, a subsystem for ,performing several tasks on particular graphs, and a theorem prover.