A snark is a non-trivial cubic graph admitting no Tait coloring.
We examine the structure of the two known snarks on 18 vertices, the Blanuša graph and the Blanuša double. By showing that one is of genus 1, the other of genus 2, we obtain maps on the torus and double torus which are not 4-colorable.
The Blanuša graphs appear also to be a counter example for the
conjecture that the orientable genus of a dot product of n Petersen graphs is n-1 (Tinsley and Watkins, 1985).
We also prove that the 6 known snarks of order 20 are all of genus 2

A. Orbanić

B. Servatius

M. Randić

T. Pisanski

English

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Mathematical Communications 9(2004), 91-103 91Blanuša double∗A.Orbanić†, T. Pisanski‡, M.Randić§and B. Servatius¶Abstract. A snark is a non-trivial cubic graph admitting no Taitcoloring. We examine the structure of the two known snarks on 18vertices, the Blanuša graph and the Blanuša double. By showing thatone is of genus 1, the other of genus 2, we obtain maps on the torus anddouble torus which are not 4-colorable. The Blanuša graphs appear alsoto be a counter example for the conjecture that the orientable genus of adot product of n Petersen graphs is n− 1 (Tinsley and Watkins, 1985).We also prove that the 6 known snarks of order 20 are all of genus 2.Key words: Blanuša snark, Blanuša double snark, dot product,embeddingsAMS subject classifications: 05C10, 05C62Received March 7, 2004 Accepted April 29, 20041. IntroductionEver since Tait [13] proved that the four-color theorem is equivalent to the statementthat every planar bridgeless cubic graph is edge 3-colorable, snarks, i.e. non-trivialcubic graphs possessing no proper edge-3-coloring, have been investigated. For anexplanation of the term non-trivial in the definition of a snark, see [7]. The smallestsnark is a graph on 10 vertices, namely the Petersen graph. It was used by Petersenin 1898, [10], but appears already in Kempe’s paper [5], see Figure 1. The Petersengraph appeared in the chemical literature as the graph that depicts a rearrangementof trigonal bipyramid complexes XY5 with five different ligands when axial ligandsbecome equatorial and equatorial ligands become axial ([9], [11]).∗The article contains topics presented at the 5th Slovenian International Conference on GraphTheory, June 22-27, 2003, Bled, Slovenia. Work supported in part by a grant from the Ministrstvoza šolstvo, znanost in šport Republike Slovenije.†Department of Mathematics, University of Ljubljana, Jadranska 19,1 000 Ljubljana, Slovenia,e-mail: Alen.Orbanic@fmf.uni-lj.si‡Department of Mathematics, University of Ljubljana, Jadranska 19,1 000 Ljubljana, Slovenia,e-mail: Tomaz.Pisanski@fmf.uni-lj.si§Dept. of Mathematics & Computer Sci., Drake University, Des Moines, IA 50311, U.S.A.,e-mail: mrandic@msn.com¶Mathematics Department, Worcester Polytechnic Institute, 100 Institute Road, Worcester,MA 01609-2280, U.S.A., e-mail: bservat@wpi.edu92 A.Orbanić, T. Pisanski, M.Randić and B. ServatiusFigure 1. Kempe’s (left), Blanuša’s (middle) and a standard rendering ofPetersen’s graphThe second snark appearing in the literature is a graph on 18 vertices, discoveredby the Croatian mathematician Danilo Blanuša, [1]. It is constructed from twocopies of the Petersen graph by a construction generalized in [3] to obtain infinitefamilies of snarks. Blanuša’s construction can be applied to the Petersen graphin two different ways yielding two snarks on 18 vertices. We call the first one theBlanuša snark and the second one the Blanuša double.Tutte conjectured that, in fact, every snark contains the Petersen graph as aminor. Robertson, Seymour and Thomas proved in [12] that Tutte’s conjecture istrue in general provided it is true for two special kinds of cubic graphs that arealmost planar. In 2001 Robertson, Sanders, Seymour and Thomas announced aproof of Tutte’s conjecture, which has not yet appeared in the literature.An interesting and still open conjecture on snarks and their embeddings intoorientable surfaces is Grünbaum’s conjecture [2] which is a generalization of the4-color theorem.Conjecture 1 [Grünbaum [2]]. Every embedding of a snark in an orientablesurface has a cycle of length 1 or 2 (a loop or a pair of parallel edges) in the dual.2. The dot productThe dot product, S1  S2, of two snarks S1 and S2 is defined in [3] as the graphobtained from S1 and S2 by removing two non-incident edges e and f from S1 andtwo adjacent vertices v and w from S2 and joining the endpoints of e and f toneighbors of u and v like in Figure 2. Performing this operation using the Petersengraph for both S1 and S2, yields, depending on the choice of the deleted edges, theBlanuša snark and the Blanuša double.Both graphs are cubic graphs on 18 vertices and are difficult to identify. Actuallyin most drawings the two are hard to distinguish. In [16], for instance, the figure ofthe same graph appears twice. In this note we show that the genus of the Blanušasnark is 1 while the genus of its double is 2. A computer search for the number of1-factors (Kekulé structures) shows that there are K = 19 Kekulé structures in theBlanuša snark and K = 20 in its double.Blanuša double 93e fuwFigure 2. The dot productFigure 3. The Blanuša snark and its double3. Genus embeddingsThe simplest non-orientable surface, in which we can embed the Petersen graph isthe projective plane (see Figure 4). The embedding in this case is pentagonal andhighly symmetric. In order to exhibit a toroidal embedding, one has to providea suitable collection of facial walks. We obtain 3 pentagonal faces, one hexagonalface and one nonagonal. We can represent this symbolically as : 536191. In thefollowing tables the numbers denote the vertices and each row represents a facedescribed as a sequence of vertices. The left half of Figure 4 is a drawing using thiscombinatorial face structure and vertex labelling.94 A.Orbanić, T. Pisanski, M.Randić and B. ServatiusFigure 4. The Petersen graph in the projective plane and in the torusThe Petersen snark on the torus:1 6 10 4 52 3 4 10 97 8 5 4 36 7 3 2 1 5 8 9 101 2 9 8 7 6In order to prove that a graph is embeddable in surface S, we have to find acombinatorial embedding, i.e. a list of faces so that the Euler characteristic is thatof S. Such a face list can be produced for the torus by the program embed[8]written by one of authors using the algorithm described in [4].To prove non-embeddability of a graph in a surface we use the fact that thereis a finite list of minimal forbidden minors for each surface (see [6]). For the planethe forbidden minors are K5 and K3,3. Unfortunately, a complete list of minimalforbidden minors for the torus is currently not known.We again used the program embed to find candidates for minimal forbiddenminors for torus embeddings: Starting with graph G one sequentially removes andcontracts edges. If a removal or a contraction of an edge produces a toroidal graphthen the operation is ignored. Otherwise the process is repeated on the obtainedgraph. When no removal or contraction of any edge is possible without obtaininga toroidal graph, we have obtained a candidate for a minimal forbidden minor. Todocument this process on the labelled graphs below, we use the following notation:n : (a b c) : i j k lmeans that the vertex labelled n in the minor was obtained by contracting alledges in a connected component on the vertices labelled a, b, c in the original graph.Vertex n is adjacent to vertices i, j, k, l in the minor. Isolated vertices of the minorare omitted.For the snarks in the sequel it appears that the obtained candidates for forbid-den minors all contain subdivisions of K3,3 and we make use of the fact that theKuratowski graph K3,3 admits exactly two essentially different unlabelled embed-dings into the torus, which are shown on fundamental polygons for the torus inFigure 5.Blanuša double 95Figure 5. Two embeddings of K3,3 into the torusThe first embedding in Figure 5 is a cellular embedding, while the other is not,since the facewalk along the decagon does not correspond to a cycle of K3,3.Theorem 1. The genus of the Blanuša snark is 1 while the genus of the Blanušadouble is 2.Proof. We find embeddings of the Blanuša snark and its double by giving thecollection of facial walks (i.e. combinatorial faces).The Blanuša snark on the torus:7 8 4 5 61 2 3 4 818 14 15 16 1711 12 13 14 181 8 7 15 14 135 9 10 3 2 617 16 12 11 10 97 6 2 1 13 12 16 1510 11 18 17 9 5 4 3Figure 6. The Blanuša snark on the torus96 A.Orbanić, T. Pisanski, M.Randić and B. ServatiusThe Blanuša double on the double torus:9 10 8 5 714 11 16 18 15 131 4 8 10 3 29 7 6 12 16 119 11 14 17 3 101 15 18 17 14 13 12 6 41 2 5 8 4 6 7 5 2 3 17 18 16 12 13 15ABABC DCD18 13123456 789 10111214151617Figure 7. Two drawings of the Blanuša double on the double torusFor the labelled Blanuša double graph in Figure 8 the depicted forbidden minorreturned by the program embed was obtained as follows.Vertices: 11, Edges: 51: (8): 7 9 22: (10): 3 1 83: (5): 9 7 24: (15): 10 11 85: (12): 7 11 106: (16): 7 10 117: (2 3 4 11): 3 8 1 5 68: (7 9): 2 7 9 49: (6 1): 8 1 310: (13 18): 5 4 611: (17 14): 4 5 6Figure 8. Blanuša double and its minimal forbidden minor for a torusThe minor in Figure 8 is not toroidal because vertices 4, 5, 6, 10, 11 would haveto be embedded in a face of K3,3. For the cellular embedding that leads to a contra-diction to Kuratowki’s theorem right away, since the face containing the specifiedvertices together with the boundary would contain another subdivision of K3,3. Ifthe specified vertices are embedded in the singular decagon of Figure 5, we observethat they are attached at two consecutive vertices of the decagon, so there exists anoncontractible curve which does not separate the vertex set 4, 5, 6, 7, 8, 10, 11, seeFigure 9, again contradicting Kuratowski’s theorem. ✷Blanuša double 97Figure 9. Cutting a torus with a non-contractible curve γOur theorem provides a counter example for the conjecture of F. C. Tinsley andJ. J. Watkins. Their conjecture was that if P is a Petersen graph and Pn stands fora dot product of n Petersen graphs, then g(Pn) = n−1 (g stands for the orientablegenus). The Blanuša snarks are both obtained as a dot product of two Petersensnarks, but their genus is different. Another counter example is Szekeres’s snark,that is obtained as a dot product of 5 Petersen graphs. From Figure 10 one caneasily find 5 disjoint subdivisions of K3,3 which cannot be embedded into a surfaceof genus 4.Figure 10. The Szekeres snark with one of the 5 disjoint subgraphs K3,3 marked bold4. The 6 snarks on 20 verticesThe labelled 6 snarks on 20 vertices are shown in Figures 11 to 16. We name the6 snarks Sn4, Sn5,. . . , Sn9. Snark Sn4 is also known as the smallest Flower snarkfrom the infinite family of flower snarks (see [3]).There are exactly 6 snarks of order 20, [16].Theorem 2. All snarks on 20 vertices have genus 2.Proof. The forbidden minors for a torus of the 6 labelled snarks are shown inFigures 11 to 16. Using similar arguments as in the proof of Theorem 1, one caneasily see that all 6 given minors are non-toroidal.98 A.Orbanić, T. Pisanski, M.Randić and B. ServatiusEmbeddings of these snarks on the double torus together with the facial walksare shown in Figures 17 to 22. The embeddings were found by program Vega [15]using an algorithm that checks all possible combinations of local rotations. ✷Sn4:Vertices: 13, Edges: 61: (6): 3 12 22: (8): 13 11 13: (9): 11 13 14: (12): 7 12 85: (15): 11 12 96: (16): 10 13 127: (17): 9 4 138: (18): 11 10 49: (19): 10 5 710: (20): 6 8 911: (1 2 7 10): 5 8 3 212: (3 4 11 5): 6 1 4 513: (13 14 ): 6 7 2 3Figure 11. Sn4 and its forbidden minorSn5:Vertices: 12, Edges: 62: (6): 3 9 113: (7): 8 2 44: (9): 9 3 105: (13): 10 12 116: (15): 10 11 127: (20): 11 10 128: (1 2 5): 3 10 99: (4 8 10): 8 4 210: (11 12 17): 7 8 6 5 411: (14 18): 5 6 7 212: (16 19): 7 6 5Vertex 3 from originalgraph removedFigure 12. Sn5 and its forbidden minorSn6:Vertices: 12, Edges: 62: (3): 12 8 93: (9): 8 10 94: (12): 8 9 105: (14): 12 11 106: (16): 11 12 107: (20): 10 11 128: (2 8): 3 2 49: (4 10): 4 3 210: (5 6 11 7 19): 5 6 7 3 411: (13 18): 5 6 712: (15 17): 7 6 5 2Vertex 1 from originalgraph removedFigure 13. Sn6 and its forbidden minorBlanuša double 99Sn7:Vertices: 11, Edges: 51: (8): 7 2 32: (9): 1 9 83: (12): 9 1 84: (14): 9 11 105: (16): 9 10 116: (20): 11 10 97: (2 4): 1 8 118: (1 3 10): 3 7 29: (5 6 11 7 19): 4 6 5 3 210: (13 18): 4 6 511: (15 17): 5 6 4 7Figure 14. Sn7 and its forbidden minorSn8:Vertices: 12, Edges: 62: (3): 3 10 83: (5): 4 9 24: (9): 3 8 105: (14): 6 9 126: (18): 7 5 117: (20): 9 6 128: (4 10): 12 4 29: (6 11 7 19): 3 11 5 710: (2 8 12): 11 2 411: (13 16): 6 12 9 1012: (15 17): 5 11 8 7Vertex 1 from originalgraph removedFigure 15. Sn8 and its forbidden minorSn9:Vertices: 11, Edges: 51: (5): 7 8 22: (9): 9 1 63: (14): 4 8 114: (18): 5 3 105: (20): 11 8 46: (1 2 4): 11 7 27: (3 10): 1 6 98: (6 11 7 19): 10 3 5 1 99: (8 12): 7 2 810: (13 16): 8 11 411: (15 17): 5 10 3 6Figure 16. Sn9 and its forbidden minor100 A.Orbanić, T. Pisanski, M.Randić and B. ServatiusList of faces:13 16 20 19 17 147 8 13 14 9 10 18 20 16 11 15 19 20 18 12 17 19 154 6 9 14 17 12 53 11 16 13 8 6 42 5 12 18 101 2 10 9 6 8 71 7 15 11 31 3 4 5 2Figure 17. Sn4 embedded into double torusList of faces:14 15 16 19 20 1811 12 13 16 15 174 6 14 18 13 12 9 7 5 83 19 16 13 18 20 17 15 14 6 7 9 102 4 8 10 9 12 111 2 11 17 20 19 31 3 10 8 51 5 7 6 4 2Figure 18. Sn5 embedded into double torusList of faces:13 16 15 14 186 7 19 20 18 14 114 9 8 12 103 10 12 11 14 15 172 3 17 20 19 16 13 5 6 11 12 81 2 8 9 5 13 18 20 17 15 16 19 71 7 6 5 9 41 4 10 3 2Figure 19. Sn6 embedded into double torusBlanuša double 101List of faces:13 16 15 14 186 7 19 20 18 14 114 5 9 8 12 11 14 15 173 9 5 6 11 12 102 4 17 20 19 16 13 10 12 81 2 8 9 31 3 10 13 18 20 17 15 16 19 71 7 6 5 4 2Figure 20. Sn7 embedded into double torusList of faces:13 18 20 17 15 1611 14 18 13 125 6 11 12 8 93 5 9 4 102 8 12 13 16 19 7 6 5 31 2 3 10 17 20 19 16 15 14 11 6 71 7 19 20 8 14 15 17 10 41 4 9 8 2Figure 21. Sn8 embedded into double torusList of faces:14 15 17 20 186 7 19 16 15 14 114 9 5 6 11 12 10 13 16 19 20 173 5 9 8 12 11 14 18 13 102 3 10 12 81 2 8 9 41 4 17 15 16 13 18 20 19 71 7 6 5 3 2Figure 22. Sn9 embedded into double torus102 A.Orbanić, T. Pisanski, M.Randić and B. ServatiusUsing the Vega program the automorphisms of the 6 snarks were calculated.Snark Sn4 (Flower snark) has 10 automorphisms and 3 vertex orbits:52 · 10 = {{4, 7, 10, 16, 17}, {6, 8, 9, 13, 14}, {1, 2, 11, 12, 19, 20, 18, 15, 5, 3}}.Snark Sn5 has 4 automorphisms and 8 vertex orbits26 · 42 = {{2, 9}, {3, 6}, {5, 8}, {11, 12}, {13, 17}, {14, 19}, {15, 16, 18, 20},{1, 4, 10, 7}}.Snark Sn6 has 4 automorphisms and12 · 23 · 43 = {{5}, {6}, {7, 11}, {3, 17}, {9, 13}, {1, 12, 19, 14}, {2, 10, 20, 15},{4, 8, 16, 18}}Snark Sn7 has a trivial automorphism group.Snark Sn8 has a trivial automorphism group.Snark Sn9 has 2 automorphisms and 11 vertex orbits:12 · 29 = {{5}, {6}, {1, 12}, {2, 8}, {3, 9}, {4, 10}, {7, 11}, {13, 17}, {14, 19},{15, 16}, {18, 20}}Figure 23. ”Knotted torus and Blanuša’s graph on it” is the title for a computermodel for a possible actual sculpture. The coordinates for the embedding of theBlanuša snark on the flat torus were produced by Tomaž Pisanski. Darko Veljansuggested that the torus should be modelled in the shape of a trefoil knot. Theactual computer drawing was produced by Marko Boben.Blanuša double 103References[1] D. Blanuša, Problem četriju boja, Glasnik Mat. Fiz. Astr. Ser II. 1(1946),31–42.[2] B.Grünbaum, Conjecture 6, in: Recent progress in combinatorics,(W.T.Tutte, Ed.), Academic Press, 1969, 343.[3] R. Isaacs, Infinite families of non-trivial trivalent graphs which are not Tait-colorable, Amer. Math. Monthly 82(1975), 221–239.[4] M. Juvan, B.Mohar, A simplified algorithm for embedding a graph into thetorus, http://www.fmf.uni-lj.si/∼mohar/Algorithms.html[5] A.B.Kempe, A memoir on the theory of mathematical form Phil. Trans. Roy.Soc. London 177(1886), 1–70.[6] B.Mohar, C.Thomassen, Graphs on Surfaces, John Hopkins UniversityPress, Baltimore and London, 2001.[7] R. Nedela, M. Škoviera, Decomposition and reductions of snarks, preprint.[8] A.Orbanić, Computer program Embed for embedding graphs into the projec-tive plane and the torus,http://www.fmf.uni-lj.si/∼mohar/Algorithms.html[9] T. Pisanski, M.Randić, Bridges between geometry and graph theory, in:Geometry at Work: A Collection of Papers Showing Applications of Geom-etry, (C.A.Gorini, Ed.), Washington, DC: Math. Assoc. Amer., 2000, 174-194,2000.[10] J. Petersen, Sur le theoréme de Tait, Intermend. Math. 15(1898), 225–227.[11] M.Randić, A systematic study of symmetry characteristic of graphs. I. Pe-tersen graph, Croat. Chem. Acta 49(1977), 643-655.[12] N.Robertson, P. Seymour, R. Thomas, Tutte’s edge coloring conjecture,J. Comb. Theory, Ser.B. 1(1997), 166-183.[13] P.G.Tait, Remarks on the colourings of maps, Proc. R. Soc. Edinburgh10(1880), 729.[14] F.C.Tinsley, J. J.Watkins, A study of snark embeddings. Graphs and ap-plications , Wiley-Intersci. 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