The orientable genus of some joins of complete graphs with large edgeless graphs

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

“…We have some partial results [5]. In particular, Conjecture 3.2 holds for g(K m + K n ) if n is even and m n. It also holds when n = 2 p + 2 for p 3 and m n − 1, and when n = 2 p + 1 for p 3 and m n + 1.…”

Local Rings with Genus Two Zero Divisor Graph

“…Hence, by the main result in [8], R = 0. If k ≥ 2, then M = x y 1 y k−1 and without loss of generality, either M 2 = x 2 or M 2 = xy 1 .…”

“…Hence, elements of 1 0 1 and 0 1 1 have degree 1 in R and do not affect its genus. The remaining graph G has a subgraph homeomorphic to K p k −1 3 and is homeomorphic to a subgraph of K p k −1 + K 3 ; by the main result in [8], we have 2 × 2 × p k = 1 4 p k − 3 when p k > 4. Similar analyses give that 2 × 3 × p k = 1 2 p k − 3 when p k > 5, that 3 × 3 × p k = 3 2 p k − 3 when p k > 9, and that 4 × p k = 1 4 p k − 3 , replacing degree two vertices with edges where necessary.…”

Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs I: Latin Square Constructions

“…More recently the first author, together with Stephens and Zha [9], determined the nonorientable genus of complete tripartite graphs K ,m,n , where ≥ m ≥ n. For n ≥ 4, the embeddings constructed for the case = m = n correspond to nonorientable hamilton cycle embeddings of K n,n .Going in the other direction, the first author and Stephens [7,8] constructed hamilton cycle embeddings of K n and used them to obtain minimum genus embeddings of K m + K n for m ≥ n − 1. In this two-part series, we extend those results to orientable surfaces for all n = 2.…”

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In an earlier paper the authors constructed a hamilton cycle embedding of K n,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. Hamilton cycle embeddings of K n,n also played a role in [8].Hamilton cycle embeddings have also been related to minimum genus embeddings in a different way. In part I, we explore a connection between orthogonal latin squares and embeddings.

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