The nonorientable genus of complete tripartite graphs

European Journal of Combinatorics, 2005

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

Discrete & Computational Geometry, 1988

Let G(m, n, k), m, n > 3, k-< min(m, n), be the graph obtained from the complete bipartite graph K,.. by deleting an arbitrary set of k independent edges, and let f(m, n, k) =[(m-2)(n-2)-k]/2. It is shown that the nonorientable genus ~(G(m, n, k)) of the graph G(m, n, k) is equal to the upper integer part off(m, n, k), except in trivial cases where f(m, n, k)-< 0 and possibly in some extreme cases, the graphs G(k,k,k) and G(k+l, k, k). These cases are also discussed, obtaining some positive and some negative results. In particular, it is shown that G(5,4,4) and G(5, 5, 5) have no nonorientable quadrilateral embedding.

arXiv: Combinatorics, 2018

In this paper, the quadrangular genus (4-genus) of the complete graph $K_p$ is shown to be $\gamma_4 (K_p) = \lceil {p(p-5)}/{8} \rceil +1$ for orientable surfaces. This means that $K_p$ is minimally embeddable in the closed orientable surface of genus $\gamma_4 (K_p)$ under the constraint that each face has length at least 4. In the most general setting, the genus of the complete graph was established by Ringel and Youngs and was mainly concerned with triangulations of surfaces. Nonetheless, since then a great deal of interest has also been generated in quadrangulations of surfaces. Hartsfield and Ringel were the first who considered minimal quadrangulations of surfaces. Sections 1--4 of this paper are essentially a reproduction of the original 1998 version as follows: Chen B., Lawrencenko S., Yang H. Determination of the 4-genus of a complete graph, submitted to Discrete Mathematics and withdrawn by S. Lawrencenko, June 1998, URL: this https URL . More discussion on this 1998 vers...

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

Discrete Mathematics, 1998

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

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

Proceedings of the American Mathematical Society, 1989

For .f a set of graphs, define the bounded chromatic number Xb(&) (resp. the bounded path chromatic number Xpfë)) to be the minimum number of colors c for which there exists a constant M such that every graph G € 2? can be vertex c-colored so that all but M vertices of G are properly colored (resp. the length of the longest monochromatic path in G is at most M). For 3? the set of toroidal graphs, Albertson and Stromquist [1] conjectured that the bounded chromatic number is 4. For any fixed g > 0 , let S'g denote the set of graphs of genus g. The Albertson-Stromquist conjecture can be extended to the conjecture that XB^g) = 4 for all g > 0. In this paper we show that 4 < Xßi^g) < 6. We also show that the bounded path chromatic number Xpi-^s) equals 4 for all g>0. Let fic(g,n)(nc{g,n)) denote the minimum / such that every graph of genus g on n vertices can be c-colored without forcing / + 1 monochromatic edges (a monochromatic path of length / + 1). We also obtain bounds for fic(g,n) and nc(g. n).

Ars Combinatoria, 2004

An orthogonal coloring of a graph G is a pair {c 1 , c 2 } of proper colorings of G, having the property that if two vertices are colored with the same color in c 1 , then they must have distinct colors in c 2. The notion of orthogonal colorings is strongly related to the notion of orthogonal Latin squares. The orthogonal chromatic number of G, denoted by Oχ(G), is the minimum

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

Discrete Mathematics, 1991

Skoviera, M., The maximum genus of graphs of diameter two, Discrete Mathematics 87 (1991) 175-180. Let G be a (finite) graph of diameter two. We prove that if G is loopless then it is upper embeddable, i.e. the maximum genus y,&G) equals [fi(G)/Z], where /3(G) = IF(G)1-IV(G)1 + 1 is the Betti number of G. For graphs with loops we show that [p(G)/21-2s yM(G) c &G)/Z] if G is vertex 2-connected, and compute the exact value of yM(G) if the vertex-connectivity of G is 1. We note that by a result of Jungerman [2] and Xuong [lo] 4-connected graphs are upper embeddable. Theorem 1. Every loopless graph of diameter two is upper embeddable.