Hamilton cycle embeddings of complete tripartite graphs and their applications (original) (raw)
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Genus embeddings for some complete tripartite graphs
Discrete Mathematics, 1976
The vu&age graph construction of Gross is extenSccf to the case where the baw graph is non-orkntably embedded. An easily applied criterion is established for determining the orientability character of the derived embedding. These methods are then applied to derive both orientable ijnd non-orientabte genus embeddings for some families of complete tripartite graphs.
The nonorientable genus of complete tripartite graphs
Journal of Combinatorial Theory, Series B, 2006
In 1976, Stahl and White conjectured that the nonorientable genus of Kl,m,n, where l ≥ m ≥ n, is (l−2)(m+n−2) 2 ¡ . The authors recently showed that the graphs K3,3,3 , K4,4,1, and K4,4,3 are counterexamples to this conjecture. Here we prove that apart from these three exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is closely related to voltage graphs.
Strong embeddings of minimum genus
Discrete Mathematics, 2010
A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph. Sporadic counterexamples to this conjecture have been known since the late 1970's. In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional to the order of these graphs), thus providing a plethora of strong counterexamples to the above conjecture. The main result yielding such counterexamples may be of independent interest.
Discrete Mathematics, 1996
The cycle double cover conjecture is equivalent to the 'pseudosurface embedding conjecture' that every 2-connected graph has a closed 2-cell embedding in some pseudosurface. The 'strong embedding conjecture' asserts that every 2-connected graph has a closed 2-cell embedding in some surface. The concern of this paper is an even stronger topological conjecture mentioned by Seymour-the 'genus strong embedding conjecture', that every bridgeless cubic graph has a closed 2-cell embedding in its minimum genus surface. A surface 2; is said to have the genus strong embedding property if every 2-connected graph for which it is the minimum-genus surface has a closed 2-cell embedding in 2;. It is well-known that the sphere has the genus strong embedding property. Negami, and Robertson and Vitray have proved that the projective plane also has the genus strong embedding property. We prove in this paper a structure theorem for minimum-genus embeddings and embeddings with the minimum number of repeated vertices and edges in their facial walks (if the strong embedding conjecture is true, then this number is zero). This structure property leads to upper bounds on the number of repeated vertices and edges in the facial walks for such embeddings of 3-connected graphs. Examples are given that show these bounds are the best possible for minimum-genus embeddings. These examples also show that the sphere and the projective plane are the only surfaces having the genus strong embedding property, thereby extending Xuong's counterexample graph for the torus. An open problem mentioned in Bender and Richmond's paper 1990 is also solved.
Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs I: Latin Square Constructions
Journal of Combinatorial Designs, 2013
In an earlier paper the authors constructed a hamilton cycle embedding of Kn,n,n in a nonorientable surface for all n ≥ 1 and then used these embeddings to determine the genus of some large families of graphs. In this two-part series, we extend those results to orientable surfaces for all n = 2. In part I, we explore a connection between orthogonal latin squares and embeddings. A product construction is presented for building pairs of orthogonal latin squares such that one member of the pair has a certain hamiltonian property. These special squares are then used to construct embeddings of the complete tripartite graph Kn,n,n on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all n ≥ 1 such that n = 2 and n = 2p for every prime p. Moreover, it is shown that the latin square construction utilized to get hamilton cycle embeddings of Kn,n,n can also be used to obtain triangulations of Kn,n,n. Part II of this series covers the case n = 2p for every prime p and applies these embeddings to obtain some genus results.
An obstruction to embedding graphs in surfaces
Discrete Mathematics, 1989
It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.
Nonorientable hamilton cycle embeddings of complete tripartite graphs
Discrete Mathematics, 2012
A cyclic construction is presented for building embeddings of the complete tripartite graph K n,n,n on a nonorientable surface such that the boundary of every face is a hamilton cycle. This construction works for several families of values of n, and we extend the result to all n with some methods of Bouchet and others. The nonorientable genus of K t,n,n,n , for t ≥ 2n, is then determined using these embeddings and a surgical method called the 'diamond sum' technique.
The maximum genus of vertex-transitive graphs
Discrete Mathematics, 1989
Non-upper-embeddable vertex-transitive graphs are characterized. A particular attention is paid to Cayley graphs. Groups for which there exists a non-upper-embeddable Cayley graph are determined+ 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers I3.V.