The cartesian product of three triangles can be embedded into a surface of genus 7

Mohar, Bojan; Pisanski, Tomazˇ; Sˇkoviera, Martin; White, Arthur T.

doi:10.1016/0012-365x(85)90197-9

Cited by 16 publications

(26 citation statements)

References 4 publications

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“…Collecting the results of [ 11, (41. [5], [9], and Section 3 of this paper, we can now state: Since there is a similar structure theorem for finite hamiltonian groups, similar asymptotic results for the orientable genus parameter can be obtained for these groups (see [7] and [lo] for the known exact results). For example, a major open case is for r = Q X Zmr X * -.…”

Section: Application To the Group Genus Parametermentioning

confidence: 56%

“…X Zm2 X Z,,, where Q is the quaternions, r 2 6, all mi are odd, and as usual mi I mi+, (1 5 i 5 r -1). We will let m , go to infinity, so we can assume that m , 2 5. Then we use Corollary 13h, Lemma 25 (if r is even), and Theorem 15 of [7] (as in Case 10 of the proof of Proposition 24; we set r, = Q X Zm, X -X Z,,) to find so that If r is o( 1, we s i result.…”

Section: Application To the Group Genus Parametermentioning

confidence: 99%

“…Thus the graph C , X C, X C3 (and the corresponding group Z , X Z , X Z3) is particularly resistant to all the constructions mentioned above. In [ 5 ] it was shown that both this graph and its corresponding group have genus at most seven, and in [ l ] it was shown that both have genus at least seven. The construction of [ 5 ] was neither surgical nor current graph (nor of the dual voltage graph form), but was ad hoc: finding a rotation scheme so as to force the desired embedding.…”

Section: Introductionmentioning

confidence: 99%

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Embeddings of cartesian products of nearly bipartite graphs

Mohar

Pisanski

White

1990

Journal of Graph Theory

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Surgical techniques are often effective in constructing genus embeddings of Cartesian products of bipartite graphs. In this paper we present a general construction that is "close" to a genus embedding for Cartesian products, where each factor is "close" to being bipartite. In specializing this to repeated Cartesian products of odd cycles, we are able to obtain asymptotic results in connection with the genus parameter for finite abelian groups.

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Section: Application To the Group Genus Parametermentioning

confidence: 56%

Section: Application To the Group Genus Parametermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Embeddings of cartesian products of nearly bipartite graphs

Mohar

Pisanski

White

1990

Journal of Graph Theory

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“…An easy computation with the MAGMA system [2] shows that the automorphism group has 148 conjugacy classes of subgroups, of orders 1, 2, 3, 4, 5, 6,7,8,9,10,12,14,16,18,20,21,24,25,32,36,40,42,48,50,60,72,80,96,100,120,125,144,168,200,240,250,336,360,480, 500, 720, 1000, 1440, 2000, 2520, 5040, 126 000 and 252 000 (with many orders repeated). We can limit our attention to those of order dividing 4|E| = 700, that is, of order 1, 2, 4, 5, 7, 10, 14, 20, 25, 50 or 100.…”

Section: Application To the Hoffman-singleton Graphmentioning

confidence: 99%

“…On the other hand, some other examples proved quite challenging, even when they were vertex-transitive. Notable cases include the Cartesian product C 3 C 3 C 3 , a 6valent graph of order 27 which took some years to deal with (see [32,4]), the 3-valent Gray graph of order 54 (see [30]), and the associated Doyle-Holt graph, a 4-valent graph of order 27 (considered 13 years ago in [30] and dealt with at last in this paper).…”

Section: Introductionmentioning

confidence: 99%

New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces

Conder

Stokes

2019

Ars Math. Contemp.

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The question of how to find the smallest genus of all embeddings of a given finite connected graph on an orientable (or non-orientable) surface has a long and interesting history. In this paper we introduce four new approaches to help answer this question, in both the orientable and non-orientable cases. One approach involves taking orbits of subgroups of the automorphism group on cycles of particular lengths in the graph as candidates for subsets of the faces of an embedding. Another uses properties of an auxiliary graph defined in terms of compatibility of these cycles. We also present two methods that make use of integer linear programming, to help determine bounds for the minimum genus, and to find minimum genus embeddings. This work was motivated by the problem of finding the minimum genus of the Hoffman-Singleton graph, and succeeded not only in solving that problem but also in answering several other open questions. The authors are very grateful to Tomaž Pisanski for suggesting that they extend their initial development of the orbit method on the Hoffman-Singleton graph to other graphs, which then led to them to develop the other methods presented here, in order to find the answers to many open questions. The authors are also grateful to the referee for some helpful suggestions about presenting their work. The first author is grateful to the N.Z. Marsden Fund for its support (grant UOA 1626), and acknowledges the use of the MAGMA system [2] for computational experiments and verification of a number of discoveries announced in this paper, as well as Sage [45] in combination with IBM CPLEX for a small number of the ILP computations. The second author acknowledges partial support from the Spanish MEC project ICWT (TIN2016-80250-R) and ARES (CONSOLIDER INGENIO 2010). E-mail addresses: m.conder@auckland.ac.nz (Marston Conder), klara.stokes@mu.ie (Klara Stokes) c b This work is licensed under https://creativecommons.org/licenses/by/4.0/

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