Nonorientable genus embedding of nearly complete bipartite graphs

TL;DR

The paper tackles determining the nonorientable genus $\widetilde{\gamma}$ of nearly complete bipartite graphs $G(m,n,k)$, where $m,n\ge 3$. It introduces extendible embeddings and a bipartite-join construction to build $\Pi(m,n,k)$ from a base graph $G(p,q,h)$ by adding copies of $G(2,2,2)$, $G(2,0,0)$, and $G(0,2,0)$, achieving $\widetilde{\gamma}(\Pi(m,n,k))=\max\{\lceil f(m,n,k)\rceil,1\}$ with $f(m,n,k)=\frac{(m-2)(n-2)-k}{2}$ for all $(m,n,k)$ outside a small set of exceptions. A key technical contribution is showing that, for $(m,n,k)$ with $(m,n,k)\notin \{(2,2,2),(3,3,3),(4,4,4),(5,5,5)\}$, every cellular embedding is extendible, enabling a constructive scheme $G(m,n,k)=G(p,q,h)\oplus G(2,2,2)^a\oplus G(2,0,0)^b\oplus G(0,2,0)^c$ to realize the bound. This leads to resolving previously open cases, such as $\widetilde{\gamma}(G(n+1,n,n))$ for even $n$ and $\widetilde{\gamma}(G(n,n,n))$ for arbitrary $n$, and provides a general, constructive method for obtaining nonorientable genus embeddings of most nearly complete bipartite graphs. The results advance the understanding of genus embeddings in bipartite graphs and yield explicit formulas for the nonorientable genus in terms of $f(m,n,k)$.

Abstract

The nearly complete bipartite graph $G(m,n,k)$ is obtained by removing $k$ independent edges from the complete bipartite graph $K_{m,n}$. In this paper, we prove that for any nearly complete bipartite graph $G(m,n,k)$ with $m, n\geq 3$, and $(m,n,k)\notin\{(5,4,4)$, $(4,5,4)$, $(5,5,5)\}$, there exists a nonorientable genus embedding $Π$ satisfying $\tildeγ(Π)=\max\{\lceil \big((m-2)(n-2)-k\big)/2\rceil, 1\}$. This embedding can be constructed by starting from an embedding of some $G(p,q,h)$ with $h\leq 6$ and $p,q\leq 7$, and then iteratively adding multiple copies of $G(2,2,2)$, $G(2,0,0)$ and $G(0,2,0)$. As a consequence, the previously unresolved nonorientable genus $\tildeγ(G(n+1,n,n))$ for even $n$ and $\tildeγ(G(n,n,n))$ for arbitrary $n$ are now determined.

Figures (9)

• Figure 1: The induced rotations at $b_n$ and $a_m$ for $k< \min\{m,n\}$.
• Figure 2: The induced rotations at $a_m$, $b_{t_1}$ and $a_{t_2}$ for $2\leq k=m<n$, where $b_{t_1}a_{t_1}\notin M_m, b_{t_{2}}a_{t_{2}}\in M_m$.
• Figure 3: The induced rotations at $b_m$, $a_{1}$ and $a_{Y}$ for $4\leq k=m=n$
• Figure 4: Inserting a handle $H$ within $(f_1,f_2)$ carrying edge set $\{x_iy_i \mid 1\leq i\leq t\}$. $H$ is an antihandle if the ordered indices $\langle i_2,i_3, \cdots,i_t\rangle=\langle t,t-1, t-2,\cdots, 2\rangle$; it is a prohandle if $\langle i_2,i_3, \cdots,i_t\rangle=\langle 2,3,\cdots, t\rangle$.
• Figure 5: Inserting a handle plus a crosscap within $\langle f_1,f_2, f_3\rangle$ carrying the edge set $E(\{b_1, b_2\}, \{a_1,a_2,a_3\})$

Theorems & Definitions (9)

• Definition 2.1
• Theorem 2.2
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Definition 3.6
• Theorem 3.7