Nonorientable genus of nearly complete bipartite 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.

European Journal of Combinatorics, 2005

In 1976, Stahl and White conjectured that the minimum nonorientable genus of K l,m,n (where

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.

Journal of Combinatorial Theory, 2007

We show that for n=4 and n⩾6, Kn has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a complete graph, Km¯+Kn=Km+n−Km, and show that for n⩾3 and m⩾n−1 its nonorientable genus

Transactions of the American Mathematical Society, 1970

With the aid of techniques developed by Edmonds, Ringel, and Youngs, it is shown that the genus of the cartesian product of the complete bipartite graph K2m,2m with itself is l + 8m2(m-1). Furthermore, let ßi" be the graph K",s and recursively define the cartesian product ßi," = ß?L x x Klfl for nä2. The genus of ß("" is shown to be 1 + 2" " 3s"(sn-4), for all n, and í even ; or for n > 1, and s = 1 or 3. The graph ßi,1' is the 1-skeleton of the «-cube, and the formula for this case gives a result familiar in the literature. Analogous results are developed for repeated cartesian products of paths and of even cycles. Introduction. In this paper a graph G is a finite 1-complex. The genus y(G) of G is the minimum genus among the genera of all compact orientable 2-manifolds in which G can be imbedded. All 2-manifolds in this paper are assumed to be compact and orientable. There are very few families of graphs for which the genus has been determined; these include the complete graphs (Ringel and Youngs [7]), the complete bipartite graphs, (Ringel [5]), and some subfamilies of the family of complete tripartite graphs (see [6] and [8]). One of the first genus formulae was developed by Ringel [4] in 1955 (and independently by Beineke and Harary [1] in 1965) when he found that the genus of the M-cube Qn is given by: y(Qn) = l+2"-3(n-4), for n ^ 2. The «-cube can be defined as a repeated cartesian product: let Qx = K2, the complete graph on two vertices, and recursively define Qn = Qn _ x x K2 for n ^ 2. In general, given two graphs Gx and G2, with vertex sets V(GX), V(G2) and edge sets E(GX), E(G2) respectively, the cartesian product Gx x G2 is formed by taking V(GX x G2) ={(ux, u2) : ux e V(GX), u2 e V(G2)} and E(GX x G2)={[(ux, u2), (vx, v2)]: ux = vx

Journal of Combinatorial Theory, Series B, 1987

A use of Euler's formula in Zaks (J. Combin. Theory Ser. B 32 (1982), 95-98 ) is replaced by an elementary argument on permutations. 0 1987 Academic Press, Inc

Discrete Mathematics, 1998

A conjecture of Robertson and Thomas on the orientable genus of graphs with a given nonorientable embedding is disproved.

Discrete Mathematics, 2009

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph Km¯+Kn with m≥n−1, the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph Km¯+Kn¯=Km,n. The orientable genus problem for Km¯+Kn with m≥n−1 seems to be more difficult, but in this paper

Canadian Journal of Mathematics, 2019

Archdeacon and Grable (1995) proved that the genus of the random graph $G\in {\mathcal{G}}_{n,p}$ is almost surely close to $pn^{2}/12$ if $p=p(n)\geqslant 3(\ln n)^{2}n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in ${\mathcal{G}}_{n_{1},n_{2},p}$. If $n_{1}\geqslant n_{2}\gg 1$, phase transitions occur for every positive integer $i$ when $p=\unicode[STIX]{x1D6E9}((n_{1}n_{2})^{-i/(2i+1)})$. A different behaviour is exhibited when one of the bipartite parts has constant size, i.e., $n_{1}\gg 1$ and $n_{2}$ is a constant. In that case, phase transitions occur when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/2})$ and when $p=\unicode[STIX]{x1D6E9}(n_{1}^{-1/3})$.

Journal of Graph Theory, 1990

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.