The nonorientable genus of joins of complete graphs with large edgeless graphs

Abstract

We show that for n = 4 and n 6, K n 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, K m + K n = K m+n − K m , and show that for n 3 and m n − 1 its nonorientable genus is (m − 2)(n − 2)/2 except when (m, n) = (4, 5). We then extend these results to find the nonorientable genus of all graphs K m + G where m |V (G)| − 1. We provide a result that applies in some cases with smaller m when G is disconnected.

The nonorientable genus of complete tripartite graphs

“…Two of us (Ellingham and Stephens) have verified (4) above for m l + 1 and (l, m) = (4, 5)-see [6] for this result, and for a discussion of previous results relevant to this conjecture.…”

“…7(a). Likewise the faces corresponding to the boundary walks(5,6,5,4,5),(1, 2, 1), (2, 3, 2), and (3, 4, 3) are shown in Fig. 7, parts (b)-(e), respectively.…”

Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph

“…In 2007, Ellingham and Stephens constructed nonorientable minimum genus embeddings of for and . The cases where have been considered in a series of articles by Korzhik [, , ], see for a summary of these.…”

Hamiltonian embeddings from triangulations

“…The recent paper by Ellingham and Stephens [5] established the existence of such embeddings in non-orientable surfaces for n = 4 and n ≥ 6. In this paper we present an entirely new construction which, by surgery on a surface triangulation of K n , generates a Hamiltonian embedding of K n on a surface of higher genus.…”