The Genus of a Graph: A Survey

Abstract

The problem of determining the genus for a graph can be dated to the Map Color Conjecture proposed by Heawood in 1890. This was implied to be a Thread Problem by Hilbert and Cohn-Vossen. The conjecture was finally established by Ringel, Youngs, and many other mathematicians. Subsequently, the genera of some special graphs with symmetry were determined. The study of genus embeddings of graphs is closely related to other invariants of a graph. Specifically, the computational complexity is dependent on the genus of the underlying graph for certain well-known NP-hard problems. In this survey, main construction techniques and certain criteria are stated in the topic of the genus of a graph. Most graphs with a known genus are listed. A new theorem is shown that the method of joint trees of a graph is reasonable. Moreover, a formal set is introduced, and related results are obtained. Although a cubic graph of Hamilton cycle is asymmetric, it is interesting that a set of associate surfaces of all its joint trees is a formal set with symmetry.

1. Introduction

In this paper, we always consider a 2-cell embedding of a connected graph on an orientable closed surface without specific explanations. Its genus is the minimum genus of a surface on which a graph is embedded. The problem of determining the genus for a graph can be dated to the Map Color Conjecture proposed by Heawood in 1890 [1].

Conjecture 1.

Set S to be an orientable (or nonorientable) surface. Let$E\left(S\right)$,$\chi \left(S\right)$, and$\lceil z\rceil$denote the Euler’s characteristic, chromatic number, and the maximum integer that is not more than z, respectively. Then

$$\chi \left(S\right)=⌈\frac{1}{2}(7+\sqrt{49-24E\left(S\right)})⌉,E\left(S\right)\ne 2.$$

If $E\left(S\right)=2$, then it is the famous Four Color Conjecture, which was established by Appel and Haken in 1976 [2]. The Map Color Conjecture was implied to be a Thread Problem by Hilbert and Cohn-Vossen (Chapter VI of [3]). This is the genus (or nonorientable genus) problem of each complete graph $K_n$ for $n\ge 3$. It is the following problem for an orientable surface.

Conjecture 2.

Let$\gamma \left(n\right)$denote the genus of$K_n$. Set$\lfloor x\rfloor$to be the smallest integer that is not less than x. Then

$$\gamma \left(n\right)=⌊\frac{(n-3)(n-4)}{12}⌋,n\ge 3.$$

The Map Color Conjecture was finally established by Ringel, Youngs, and many other mathematicians in 1968 [4]. The proof of the conjecture was important progress in the field of topological graph theory. New methods were proposed to determine the genera of graphs, especially graphs with the high symmetry, during and after this period. Moreover, Thomassen proved that determining the genus of a graph is NP-complete [5]. The study of genus embeddings of graphs is closely related to the study of geometric realizations of set systems. Especially, the computational complexity is dependent on the genus of the underlying graph for the well-known NP-hard problems, which are INDEPENDENT SET, VERTEX COVER, and DOMINATING SET. This paper mainly reviews some methods of constructing the genus embeddings of some special graphs with symmetry. Classical criteria and certain generalizations are presented. Most graphs with known genera are listed, which updates the table provided by Sthal [6]. Furthermore, a new conclusion is proved, which implies that joint trees of a graph are reasonable. A formal set is introduced, and related results are provided. Specifically, although a cubic graph of Hamilton cycle is asymmetric, the set of associate surfaces of all its joint trees is a special formal set with symmetry.

We organize this rest of paper as follows. In Section 2, we sketch some concepts and conclusions. In Section 3, we review main methods to determine the genus of a graph. Moreover, a new result is provided that implies that joint trees of a graph are reasonable. We give a review of classical planarity criteria and a generalization in Section 4. In Section 5, a formal set is proposed and related results are provided. Moreover, we prove that a set of all associate surfaces for a Hamiltonian cubic graph is a special formal set. In addition, most of the graphs with known genera are listed in Appendix A.

2. Preliminaries

In this section, basic concepts are sketched. These are mainly taken from [7]. The other undefined terms can be found in [8].

For a simple graph $G=(V,E)$, the graph $\overline{G}=(V,\overline{E})$ is called the complement graph of G such that for each $uv\in \overline{E}$ if and only if $uv\notin E$. Given a set V of intervals on a line, regard each interval as a vertex. If $u\cap v\ne \varnothing$ and neither $u\subseteq v$ nor $v\subseteq u$ for each $uv\in E$, then $G=(V,E)$ is called an overlap graph where ∅ is a set that does not contain any element.

Given two graphs $G_1$ and $G_2$, the union $G_1\bigcup G_2$ is a graph. Here $V(G_1\bigcup G_2)=V\left(G_1\right)\bigcup V\left(G_2\right)$, $E(G_1\bigcup G_2)=E\left(G_1\right)\bigcup E\left(G_2\right)$. The concept of union can be generalized to a finite collection of graphs. If all the graphs are the same, we have the notation $nG={\bigcup }_{i=1}^n\left(G\right)$. The join $G=G_1+G_2$ is a graph where $V\left(G\right)=V\left(G_1\right)\bigcup V\left(G_2\right)$ and $E\left(G\right)=E\left(G_1\right)\bigcup E\left(G_2\right)\bigcup \left\{uv\right|\forall u\in V\left(G_1\right),\forall v\in V\left(G_2\right)\}$. If $G_1$ and $G_2$ are empty graphs, then G is a complete bipartite graph. Therefore, $\overline{K_m}+\overline{K_n}=K_{m,n}$, where $K_n$ denotes a complete graph of order n. The Cartesian product $G_1\times G_2$ is a graph where $V(G_1\times G_2)=V\left(G_1\right)\times V\left(G_2\right)$ and $E(G_1\times G_2)=\left\{(u_1,v_1)(u_2,v_2)\right|\mathrm{either}{u}_1=u_2\mathrm{and}{v}_1{v}_2\in E\left(G_2\right)\mathrm{or}{v}_1=v_2\mathrm{and}{u}_1{u}_2\in E\left(G_1\right)\}$. The composition (or lexicographic product), denoted by $G=G_1\left[G_2\right]$ is also a graph where $V\left(G\right)=V\left(G_1\right)\times V\left(G_2\right)$ and $E\left(G\right)=\left\{(v_1,w_1)(v_2,w_2)\right|v_1=v_2$ and $w_1{w}_2\in E\left(G_2\right)\mathrm{or}{v}_1{v}_2\in E\left(G_1\right)\}$. If H is a subgraph both of $G_1$ and of $G_2$, then an amalgamation $G_1{\vee }_H{G}_2$ of $G_1$ and $G_2$ along H is the graph by identifying a copy of H contained in $G_1$ with a copy of H contained in $G_2$. The tensor product (Kronecker product or conjunction) $G_1⨂G_2$ is the graph where $V(G_1⨂G_2)=V\left(G_1\right)\times V\left(G_2\right)$ and $E(G_1⨂G_2)=\left\{(v_1,w_1)(v_2,w_2)\right|v_1{v}_2\in E\left(G_1\right)\mathrm{and}{w}_1{w}_2\in E\left(G_2\right)\}$.

For a given graph $G=(V,E)$ and a vertex $v\in V\left(G\right)$, a rotation $\sigma \left(v\right)$ is a cyclic permutation of edges that are incident with v. Assign a rotation to each vertex of $V\left(G\right)$, then obtain a rotation system of G. Edmonds found the following result [9]. Youngs provided the result of the first proof published [10]. The idea of the bijection can be found in earlier work about embeddings [11,12].

Proposition 1.

There exists a bijection between the rotation systems and its orientable embeddings for a graph.

The polygon representation of a surface was found in the argument of the classification of closed surfaces. Let $\mathcal{S}$ denote the set of all surfaces. Each surface is equivalent to one and only one of canonical forms by using the equivalence “∼” defined on $\mathcal{S}$ with the following operation properties [13]:

Op1.

$A\sim Aa{a}^{-}$, where $A\in \mathcal{S}$ and $a\notin A$;

Op2.

$AabB{b}^{-}{a}^{-}\sim AcB{c}^{-}=A{c}^{-}Bc$, where $AB\in \mathcal{S}$ and $a,b,c\notin AB$ for linear sequences A and B;

Op3.

$AaBbC{a}^{-}D{b}^{-}E\sim ADCBEab{a}^{-}{b}^{-}$, where $A,B,C,D$ and E are linear sequences and $AaBbC{a}^{-}$$D{b}^{-}E$$\in \mathcal{S}$.

Let $\mathcal{S}$ denote the set of all surfaces. Let T be a spanning tree of G, and let $G_{\sigma }$ be an embedding. Given a rotation system $\sigma$, splitting each cotree edge b into two semi-edges b and $b^{-}$, we get a joint tree ${\tilde{T}}_{\sigma }$ [13]. The associate surface ( or embedding surface ) $S_{\sigma }$ of ${\tilde{T}}_{\sigma }$ is a polyhegon containing all letters of semi-edges. It is obvious that there is one and only one joint tree ${\tilde{T}}_{\sigma }$ for a given rotation system of $G_{\sigma }$. Is the associate surface $S_{\sigma }$ the surface that $G_{\sigma }$ embeds on? The answer is affirmative. A proof is given in the following section. Thus, determining the genus of G is transformed into determining the genus of the set of its associate surfaces.

3. Methods for Determining the Genus of a Graph

In this section, we review methods used successfully to determine the genera of some special graphs. The reader is referred to the survey in [6]. Furthermore, we provide a new result related to the joint tree of a graph.

There exists several main methods to settle the genus of a graph. The first approach is the direct construction using rotation systems. Heffter first used it to get the genera of some $K_n$, and then Ringel calculated the genera of $K_{m,n}$ and certain $K_n$ [4,14]. Subsequently, Ringel and Youngs obtained the genus of $K_{n,n,n}$ [15]. White calculated the genera of several families of complete tripartite graphs, including $K_{mn,n,n}$ [16].

The second method is a current graph denoted by $(H,\mathsf{\Gamma },\eta )$. Here, H is an embedding, the finite group $\mathsf{\Gamma }$ has an identity e, and $\eta :H\to \mathsf{\Gamma }-e$ is a function such that ${\left(\eta \left(x\right)\right)}^{-1}=\eta \left(x^{-1}\right)$ for any $x\in E\left(G\right)$; $\eta \left(x\right)$ is regarded as a current. It was created by Gustin [17] and developed by Youngs. Ringel and Youngs used it to compute the genera of some $K_n$. Jacques gave a general exposition for Cayley graphs [18]. Finally, Gross and Alpert unified all previously definitions of current graphs into one [19,20].

The third method is a voltage graph. It is a triple $(H,\mathsf{\Gamma },\eta )$ with an embedding H, a finite group $\mathsf{\Gamma }$ of identity e, and a function $\eta :H\to \mathsf{\Gamma }$ such that ${\left(\eta \left(x\right)\right)}^{-1}=\eta \left(x^{-1}\right)$ for all $x\in H$. Here, $\eta \left(x\right)$ is a voltage. Its derived graph $H\times \mathsf{\Gamma }$ has vertex set $V\left(H\right)\times \mathsf{\Gamma }$. Two vertices $(u_1,{\xi }_1)$ and $(u_2,{\xi }_2)$ in $H\times \mathsf{\Gamma }$ are adjacent if and only if $u_1$ and $u_2$ are adjacent in H such that $\eta \left(u_1{u}_2\right)={\xi }_1^{-1}{\xi }_2$. A voltage graph was developed by Gross in 1974 [21]. It is interpreted in the context of branched covering spaces and regarded as a dual of a current graph [19,20]. Gross and Tucker extended it to nonregular coverings via the permutation voltage [22].

The fourth method is a transition graph denoted by $\mathcal{G}=(D,\mathcal{T},\lambda ,\alpha )$, which was introduced by Ellingham et al. in 2006 [23]. Here,

(1)

D is a diagraph where both the indegree and the outdegree are equal to 2 for any $u\in V\left(D\right)$;

(2)

$\mathcal{T}=\{C_1,\cdots$$,C_m\}$, where each $C_i$ is a closed trails for $1\le i\le m$ and where $E\left(D\right)={\displaystyle \bigcup _{i=1}^m}{C}_i$ and where $C_i$ and $C_j$ don’t have a common edge for $i\ne j$;

(3)

an ordering $C_i\to C_j$ of the $C_i$ and $C_j$ incident with u for each $u\in V\left(D\right)$;

(4)

a function $\lambda$: $V\to \{-1,+1\}$;

(5)

a function $\alpha$: $V\to \mathsf{\Gamma }$, where (usually infinite group) $\mathsf{\Gamma }$ is a voltage group and $\alpha$ is a voltage assignment.

This is equivalent to a voltage graph. Vertices and edges of a voltage graph correspond to vertices and edges of its derived graph, respectively. In case of a transition graph and the derived graph, vertices correspond to edges, and edges correspond to consecutive pairs of edges in the local rotations. The number and sizes of the derived faces can be easily determined.

The fifth method is a surgery, which is a scissors-and-paste approach to obtain a genus embedding of a graph. It was used to calculate the genus of the n-cube $Q_n$ by Ringel, and Beineke and Harary [24,25]. White generalized this approach to establish many genus formulae for Cartesian, lexicographic, and strong tensor product graphs [16,26]. Alpert used it to obtain some results for amalgamations [27]. Ma and Ren provided the genus of $C_m\times K_n$ by two surgical constructions where $n=4$ and $m\ge 12$, or $n\ge 5$ and $m\ge 6n-13$. Lv and Chen provided the genus of $K_{n,n,1}$ by a handle-inserting operation for odd n.

Bouchet introduced a special surgical technique called a diamond sum to give a new proof of the genera of complete bipartite graphs in 1978 [28]. A primal version was applied by Mohar et al. [29,30]. Mohar and Thomassen gave a primal form of Bouchet’s proof and used the definition [31]. A general form was given by Kawarabayashi et al. [32]. Let ${\mathsf{\Psi }}_1:G\to S_1$ and ${\mathsf{\Psi }}_2:H\to S_2$ be two embeddings of G and H into the surfaces $S_1$ and $S_2$, respectively. Set $x\in V\left(G\right)$, $y\in V\left(H\right)$, adjacent to m vertices $x_1,x_2,\cdots$$,x_m\in V\left(G\right)$ and $y_m,y_{m-1},\cdots$$,y_1\in V\left(H\right)$, respectively, in the clockwise rotation. Set $D_1$ to be a closed disk which contains in a small neighborhood of the star $st\left(x\right)=\{x{x}_1,x{x}_2,\cdots$$,x{x}_m\}$. Here, it contains $st\left(x\right)$ and intersects G only at $x_1,x_2,\cdots$$,x_m$. Similarly, choose $D_2$ in a small neighborhood of the star $\left\{y\right\}\bigcup \{y{y}_1,y{y}_2,\cdots$$,y{y}_m\}$. Delete the interior of $D_i$ from $S_i$ for $1\le i\le 2$, identify their boundaries of $S_i\backslash D_i$, and then get an embedding $\mathsf{\Psi }$ of a new graph U in the surface $S_1\circ S_2$. Here, $S_1\circ S_2$ is the disk sum of $S_1$ and $S_2$; U is obtained from $G\backslash \left\{x\right\}$ and $H\backslash \left\{y\right\}$ by identifying $x_i$ with $y_i$ for $i=1,2,\cdots$$,m.$ The operations on the graph and the embedding are, respectively, called the diamond sum of graphs and the diamond sums of embeddings, denoted by

$$(G,u)♢(H,v)$$

and

$${\mathsf{\Psi }}_1(G,u)♢{\mathsf{\Psi }}_2(H,v).$$

The sixth method is generative n-valuations introduced by Bouchet in 1976 [33]. Let H be a Eulerian graph and $H_{\sigma }$ is an embedding of H. If the boundary of each face for $H_{\sigma }$ is a triangle, then $H_{\sigma }$ is called an even triangulation. Assign to each triangle of $H_{\sigma }$ an element of $Z_n$ and obtain a map $\eta$ called an n-valuation of an even triangulation. Set $u\in V\left(H\right)$ and suppose $\sigma \left(u\right)$ is a rotation at u conformal with $H_{\sigma }$. The rotation induces a cyclic order of all the triangles of $H_{\sigma }$ incident to u, denoted by ${\mathsf{\Delta }}_1,{\mathsf{\Delta }}_2,\cdots$$,{\mathsf{\Delta }}_{2l}$. Set

$$\lambda \left(u\right)=\sum _{i=1}^{2l}{(-1)}^i\eta \left({\mathsf{\Delta }}_i\right).$$

If $\eta$ is an n-valuation such that $\lambda \left(u\right)$ is a generator for any $u\in V\left(H\right)$, then $\eta$ is called a generative n-valuation of $H_{\sigma }$. This approach was applied in [33,34].

The seventh method is a joint tree, which was introduced by Liu [13]. Wan et al. also verified the genera of complete bipartite graphs by using joint trees, the paper for which was posted on Science Online in 2012 [35]. Shao and Liu used this technique to obtain genus embeddings of complete bipartite graphs $K_{n,n,l}$ for $l\ge n\ge 2$ [36]. Shao et al. obtained the genera of other graphs [37,38].

We prove the following result. The result explains that the associate surface of a joint tree ${\tilde{T}}_{\sigma }$ is the surface that it is embedded on for any embedding $G_{\sigma }$ of G. The result holds for any nonorientable embedding by applying a similar argument.

Theorem 1.

Set T to be a spanning tree of a connected graph G. Let $G_{\sigma }$ and ${\tilde{T}}_{\sigma }$ be the embedding and the joint tree for any rotation system σ of G, respectively. Set $S_{\sigma }$ to be the associate surface of ${\tilde{T}}_{\sigma }$. Then $S_{\sigma }$ is the surface that $G_{\sigma }$ embeds on.

Proof. 

If $G=T$, then $S_{\sigma }$ is a sphere. The result obviously holds. Suppose that G has m cotree-edges denoted by $x_1,x_2,\cdots$$,x_m$, where $m\ge 1$. We verify the result by induction on the edge number n of T.

Now consider the case $n=0$. Suppose that $\sigma =(x_{i_1}{x}_{i_2}{x}_{i_3}\cdots x_{i_{2m}})$, where each $x_j$ occurs twice in $\sigma$ for $1\le j\le m$. Let S be the surface of $G_{\sigma }$ embeds on. Then S is formed by identifying edges with the same letters of the polygon with the symbol $x_{i_1}^{{\epsilon }_1}{x}_{i_2}^{{\epsilon }_2}{x}_{i_3}^{{\epsilon }_3}\cdots x_{i_{2m}}^{{\epsilon }_{2m}}$. Here, $x_j$ and $x_j^{-}$ occur once on the polygon for $1\le j\le m$. The expression ($x_{i_1}^{{\epsilon }_1}{x}_{i_2}^{{\epsilon }_2}{x}_{i_3}^{{\epsilon }_3}\cdots x_{i_{2m}}^{{\epsilon }_{2m}}$) is in fact the associate surface $S_{\sigma }$ of the joint tree ${\tilde{T}}_{\sigma }$. The result therefore holds.

Assume that the result holds for any integers less than n$(n\ge 1)$. Next we verify the case for n. Because T is a tree, there exists a vertex u such that u is incident with one and only one edge in T. Without loss of generality, set $e_1=uv$, ${\sigma }_u=(e_1,x_{i_1},x_{i_2},\cdots$$,x_{i_r})$ and ${\sigma }_v=(e_1,L_1,e_2,L_2,\cdots$$,e_k,L_k)$, where each $L_j$ is either empty or $L_j$ consists of certain cotree edges for $k\ge 1$, $1\le j\le k$, $1\le i_l\le m$ for each $1\le l\le r$ and some $1\le r\le 2m$. Here, each $e_j$ is a tree edge for $1\le j\le k$.

Set $H=G·e_1$. Therefore, $T^{\prime }=T·e_1$ is a spanning tree of H. Denote the new vertex with w by contracting the edge $e_1$ and keep the labels of all other verties. Let $\tau$ be the rotation system of H by setting ${\tau }_w=(x_{i_1},x_{i_2},\cdots$$,x_{i_r},L_1,e_2,L_2,\cdots$$,e_k,L_k)$ and ${\tau }_a={\sigma }_a$ for $a\ne w$ and $a\in V\left(H\right)$. According to the definition of the associate surface of a joint tree,

$$S_{\tau }=S_{\sigma }$$

(1)

where $S_{\tau }$ and $S_{\sigma }$ are the associate surfaces of the joint tree ${\tilde{T}}_{\tau }^{\prime }$ of H and the joint tree ${\tilde{T}}_{\sigma }$ of G, respectively. It is clear that

$$V\left(H_{\tau }\right)=V\left(G_{\sigma }\right)-1,E\left(H_{\tau }\right)=E\left(G_{\sigma }\right)-1\mathrm{and}\varphi \left(G_{\sigma }\right)=\varphi \left(H_{\tau }\right).$$

Using the Euler–Ponincaré formula,

$$V\left(G_{\sigma }\right)-E\left(G_{\sigma }\right)+\varphi \left(G_{\sigma }\right)=2-2\gamma \left(G_{\sigma }\right)$$

and

$$V\left(H_{\tau }\right)-E\left(H_{\tau }\right)+\varphi \left(H_{\tau }\right)=2-2\gamma \left(H_{\tau }\right).$$

Thus

$$\gamma \left(G_{\tau }\right)=\gamma \left(H_{\tau }\right).$$

By induction assumption, $S_{\sigma }$ is the surface that $G_{\sigma }$ embeds on. Therefore, $S_{\sigma }$ is the surface that $G_{\sigma }$ embeds on by applying Equation $\left(1\right)$.

Hence the result holds by induction. □

Moreover, Conder and Stokes introduced methods that are the subgroup orbit, the independence number, and the use of integer linear programming [39]. They used these techniques to get the genera of $C_3\times C_3\times C_3$ and the Gray graph, etc. In addition, Mohar et al. and Brin and Squier obtained the genus of $C_3\times C_3\times C_3$ [40,41]. Marušič et al. supplied the genus of the Gray graph [42].

In addition, some of these methods can be combined together to calculate the genus of a graph. Surgery can be used to augment embeddings derived from other methods, as in the additional adjacency constructions of Ringel and Youngs [4]. White combined the voltage graph theory and a surgery. If a voltage–current group is Abelian, then Archdeacon supplied an approach for simultaneously assigning a voltage and a current on an embedded graph [43]. He used it to obtain the genera of $K_{n,n,n}$ for $n\ge 2$. Most of graphs with known genera are listed in Appendix A, where Stahl’s table is updated.

4. Planarity Criteria and Generalizations

It is a very important problem to characterize graphs embedded on surfaces, which can be used to determine the genus of a graph. In this section, we review several classical planarity criteria and certain generalizations.

Kuratowski proved the following famous result to characterize those graphs that embed on the plane in 1930 [44].

Proposition 2.

It is planar if and only if a graph does not contain a subgraph homomorphic to $K_5$ and $K_{3,3}$.

If G can be formed from a subgraph of H by contracting edges, G is a minor of H. Wagner rephrased the result above by using the minor as follows [45].

Proposition 3.

It is planar if and only if a graph does not contain $K_5$ and $K_{3,3}$ as a minor.

A cocycle is a minimal edge cut. If there exists a function $\lambda :E\left(H\right)\to E\left(H^{\ast }\right)$ such that each cycle of H is a cocycle of $H^{\ast }$, then $H^{\ast }$ is an algebraic dual of H. Whitney found the following result [46].

Proposition 4.

A graph is planar if and only if it has an algebraic dual.

If $H^{\ast }$ is an algebraic dual of H, then there is a planar embedding of H such that $H^{\ast }$ is the geometric dual. MacLane characterized the planarity by a certain abstract property of its cycle space [47,48].

Proposition 5.

A 2-connected graph G is planar if and only if there is a family $\mathcal{C}$ of $\beta \left(G\right)$ cycles such that each $e\in E\left(G\right)$ occurs either once or twice in $\mathcal{C}$.

Set H to be a graph of $\delta \left(H\right)\ge 3$ and set S to be a surface. If H is not embedded on S and $G-x$ is embedded on S for any $x\in E\left(H\right)$, then H is a topological obstruction for S. If H with each edge x contracted can be embedded on S for a topological obstruction H, then it is a minor order obstruction of S. Vollmerhaus, Bodendiek and Wagner, and Robertson and Seymour independently proved the following result, which is a generalization of the Kuratowski theorem, to an arbitrary surface [49,50,51].

Proposition 6.

There is a finite set of obstructions (topological or minor order) for every surface of fixed genus.

However, it is very difficult to find the complete list of obstructions for a surface of genus $g\ge 1$. Gagarina et al. obtained the following result [52].

Proposition 7.

Suppose that a graph G does not contain $K_{3,3}$ as a minor. G is toroidal if and only if it has no $G_i$ as a minor in Figure 1 for $1\le i\le 4$.

5. Formal Sets

In this section, we introduce a formal set that is useful to compute the genus of a graph. Specifically, if G is a cubic graph with a Hamilton cycle, then the set of all its associate surfaces is a special formal set.

A formal sequence $\mathcal{A}$ is a sequence consisting of lowercase letters and uppercase letters. Let n and $m_j$ be positive integers and let $x_l$ be distinct letters for $1\le l\le n$ and $1\le j\le 2n$. A formal set $\mathcal{F}$ of size n is a formal sequence consisting of $x_l^{ϵ}$, $X_{l,s_l}$, and $Y_{l,s_{n+l}}$ for $ϵ\in \{+,-\}$, $1\le l\le n$, $1\le s_l\le m_l$ and $1\le s_{l+n}\le m_{l+n}$ satisfying the following conditions:

(1)

If $m_l=1$, then $x_l$ occurs once on $\mathcal{F}$, otherwise $X_{l,s_l}$ occurs once for each $1\le s_l\le m_l$;

(2)

If $m_{l+n}=1$, then $x_l^{-}$ occurs once on $\mathcal{F}$, otherwise $Y_{l,s_{l+n}}$ occurs once for each $1\le s_{l+n}\le m_{l+n}$

where

$$\left\{\begin{array}{c}{X}_{l,s_l}\in \{x_l,\varnothing \},{\displaystyle \bigcup _{s_l=1}^{m_l}}{X}_{l,s_l}=x_l,X_{l,p}\bigcap X_{l,q}=\varnothing ,\hfill \\ \mathrm{if}1\le s_l\le m_l\mathrm{and}\mathrm{any}1\le p\ne q\le m_l;\hfill \\ Y_{l,s_{n+l}}\in \{x_l^{-},\varnothing \},{\displaystyle \bigcup _{s_{n+l}=1}^{m_{n+l}}}{Y}_{l,s_{n+l}}=x_l^{-},Y_{l,p}\bigcap Y_{l,q}=\varnothing ,\hfill \\ \mathrm{if}1\le s_{n+l}\le m_{n+l}\mathrm{and}\mathrm{any}1\le p\ne q\le m_{n+l}.\hfill \end{array}\right.$$

Here, $m_l$ and $m_{n+l}$ are called valencies of $x_l$ and $x_l^{-}$, denoted by $d\left(x_l\right)$ and $d\left(x_l^{-}\right)$, respectively. Specifically, if $m_l=2$ for $1\le l\le n$, then let $X_{l,1}=X_l$ and let $X_{l,2}={\overline{X}}_l$. If $m_{l+n}=2$ for $1\le l\le n$, then let $Y_{l,1}=Y_l$ and let $Y_{l,2}={\overline{Y}}_l$.

Let $\mathcal{A},\mathcal{B}$, and $\mathcal{C}$ be formal sequences. A formal set has no difference if one uses the following operations:

(1)

Exchange $\mathcal{A}$ and $\mathcal{B}$ for a formal set $\mathcal{A}\mathcal{B}$;

(2)

Exchange $X_{l,p}$ and $X_{l,q}$ or exchange $Y_{l,p}$ and $Y_{l,q}$ for $1\le l\le n$ and $p\ne q$;

(3)

Replace each $x_l$ by $z^{ϵ}$ and simultaneously replace each $x_l^{-}$ by $z^{-ϵ}$ for $z\notin \mathcal{F}$ and $ϵ\in \{+,-\}$;

(4)

Replace $X_{l,p}$ (or $Y_{l,p}$) by Z for $Z\notin \mathcal{F}$, $Z\in \{x_l,\varnothing \}$ (or $Z\in \{x_l^{-},\varnothing \}$), $1\le l\le n$ and $1\le p\le m_l$ (or $1\le p\le m_{n+l})$.

If a formal set $\mathcal{F}=x{Z}_1{Z}_2\cdots Z_{2n}{x}^{-}{\overline{Z}}_{2n}\cdots {\overline{Z}}_2{\overline{Z}}_1$, where $Z_1,Z_2,\cdots$$,Z_{2n}$ is a permutation of $X_i$ and $Y_i$ for $1\le i\le n$, then $\mathcal{F}$ is called a cubic basic set.

In fact, a formal set $\mathcal{F}$ is a set of orientable surfaces. A surface in $\mathcal{F}$ is obtained by replacing an uppercase letter by the associated letter according to its definition. Then $\mathcal{F}$ contains ${\displaystyle \prod _{j=1}^{2n}}{m}_j$ orientable surfaces; ${\dot{U}}_{x_l}=\{x_l,X_{l,s_l}|1\le s_l\le m_l\}$ and ${\dot{U}}_{x_l^{-}}=\{x_l^{-},Y_{l,s_{n+l}}|1\le s_{n+l}\le m_{n+l}\}$ are called siblings of $x_l$ and $x_l^{-}$, respectively. For convenience, let $\mathsf{\Omega }$ and ${\mathsf{\Omega }}_n$ denote the family of all formal sets and of all formal sets of size n, respectively.

Example 1.

Let $\mathcal{F}$$=X_{2,1}{x}_1{Y}_{1,1}{X}_{2,2}{x}_2^{-}{X}_{2,3}{Y}_{1,2}\in \mathsf{\Omega }$ where $Y_{1,i}\in \{x_1^{-},\varnothing \}$ for $1\le i\le 2$, $X_{2,j}\in \{x_2,\varnothing \}$ for $1\le j\le 3$,$Y_{1,1}\bigcup Y_{1,2}=x_1^{-}$, $Y_{1,1}\bigcap Y_{1,2}=\varnothing$, ${\displaystyle \bigcup _{j=1}^3}{X}_{1,j}=x_2$, ${\displaystyle \bigcap _{j=1}^3}{X}_{1,j}=\varnothing$. Then $\mathcal{F}$ consists of the following surfaces:

$x_2{x}_1{x}_1^{-}{x}_2^{-}$$x_1{x}_1^{-}{x}_2{x}_2^{-}$$x_1{x}_1^{-}{x}_2^{-}{x}_2$$x_2{x}_1{x}_2^{-}{x}_1^{-}$$x_1{x}_2{x}_2^{-}{x}_1^{-}$$x_1{x}_2^{-}{x}_2{x}_1^{-}.$

Given a formal set $\mathcal{F}$, its genus denoted by $\gamma \left(\mathcal{F}\right)$ is the minimum genus of surfaces contained in $\mathcal{F}$. If it contains a surface of genus 0, then it is planar. In order to study the planarity, we construct its accompanying graph $Acc\left(\mathcal{F}\right)$. Put each letter on the real line according to the order in $\mathcal{F}$. We regard $[x_l,x_l^{-}]$ (or $[x_l^{-},x_l]$), $[x_l,Y_{l,s_{n+l}}]$ (or $[Y_{l,s_{n+l}},x_l]$), $[X_{l,s_l},x_l^{-}]$ (or $[x_l^{-},X_{l,s_l}]$), and $[X_{l,s_l},Y_{l,s_{n+l}}]$ (or $[Y_{l,s_{n+l}},X_{l,s_l}]$) as intervals, denoted by $u_{l,0,0}$, $u_{l,0,s_{n+l}}$, $u_{l,s_l,0}$, and $u_{l,s_l,s_{n+l}}$ for $1\le l\le n$, $1\le s_l\le m_l$, and $1\le s_{n+l}\le m_{n+l}$, respectively; $Acc\left(\mathcal{A}\right)$ is the associated overlap graph with vertices $u_{l,j,k}$ and edges such that $u_{p,j,k}$ is adjacent to $u_{q,j_1,k_1}$ if and only if $u_{p,j,k}\cap u_{q,j_1,k_1}\ne \varnothing$ (empty set) and neither $u_{p,j,k}\subseteq u_{q,j_1,k_1}$ nor $u_{q,j_1,k_1}\subseteq u_{p,j,k}$ for $1\le l,p,q\le n$, $0\le j\le m_p$, $0\le j_1\le m_q$, $0\le k\le m_{n+p}$ and $0\le k_1\le m_{n+q}$.

Theorem 2.

For $\mathcal{A}\in {\mathsf{\Omega }}_n$, $\mathcal{F}$ is planar if and only if the accompanying graph Acc($\mathcal{F}$) contains an independent set I such that $u_{l,j,k}\in I$ for each $1\le l\le n$ and some $j,k$ with $j=0$ for $d\left(x_l\right)=1$, $k=0$ for $d\left(x_l^{-}\right)=1$, $1\le j\le m_l$, and $1\le k\le m_{n+l}$ for others where $d\left(x_l\right)=m_l$ and $d\left(x_l^{-}\right)=m_{n+l}$.

Proof. 

Suppose that $\mathcal{F}$ is planar. Then there exists a surface $S\in \mathcal{F}$ such that $\gamma \left(S\right)=0$, where S consists of $x_l$ and $x_l^{-}$ for $1\le l\le n$. Consider the intervals $[x_l,x_l^{-}]$, denoted by $u_l$ for notational convenience, which are determined by their orders on S. It is obvious that $u_p\cap u_q=\varnothing$, $u_p\subseteq u_q$ or $u_q\subseteq u_p$ for $p\ne q$. and $1\le p,q\le n$. Since $S\in \mathcal{F}$, there exists $u_{l,j_l,k_l}$ such that $u_{l,j_l,k_l}=u_l$ for each $1\le l\le n$ and some $j_l,k_l$ with $j_l=0$ for $d\left(x_l\right)=1$, $k_l=0$ for $d\left(x_l^{-}\right)=1$, $1\le j_l\le m_l$ and $1\le k_l\le m_{n+l}$ for others where $d\left(x_l\right)=m_l$ and $d\left(x_l^{-}\right)=m_{n+l}$. Thus, $I=\left\{u_{l,j_l,k_l}\right|1\le l\le n\}$ is an independent set of $Acc\left(\mathcal{F}\right)$.

Conversely, suppose that I is an independent set of $Acc\left(\mathcal{F}\right)$ such that $u_{l,j_l,k_l}=u_l$ for each $1\le l\le n$ and some $j_l,k_l$ with $j_l=0$ for $d\left(x_l\right)=1$, $k_l=0$ for $d\left(x_l^{-}\right)=1$, $1\le j_l\le m_l$ and $1\le k_l\le m_{n+l}$ for others where $d\left(x_l\right)=m_l$ and $d\left(x_l^{-}\right)=m_{n+l}$. Let S be the the associated surface of $\mathcal{F}$ such that

$$\left\{\begin{array}{c}{X}_{l,j_l}=x_l,\mathrm{if}{j}_l\ge 1;\hfill \\ Y_{l,k_l}=x_l^{-},\mathrm{if}{k}_l\ge 1;\hfill \\ x\in S,\mathrm{if}d\left(x\right)=1\mathrm{for}x\in \{x_l,x_l^{-}|1\le l\le n\}.\hfill \end{array}\right.$$

It is obvious that $\gamma \left(S\right)=0$. □

If $xy,xZ$, or $XZ\in \mathcal{A}$, for $y,Z\in {\dot{U}}_{x^{-}}$ and $X\in {\dot{U}}_x$, then x is an eliminable letter.

Lemma 1.

Let $\mathcal{A}\in {\mathsf{\Omega }}_n$, let $x_p$ be an eliminable letter, and let ${\dot{U}}_{x_p,x_p^{-}}={\dot{U}}_{x_p}\bigcup {\dot{U}}_{x_p^{-}}$ for some $1\le p\le n$. Then $\gamma ($$\mathcal{A}$)=$\gamma ($$\mathcal{B}$) where $\mathcal{B}=\mathcal{A}\backslash {\dot{U}}_{x_p,x_p^{-}}$.

Proof. 

Without loss of generality, suppose that $\mathcal{A}={\mathcal{A}}_1{X}_{p,j}{Y}_{p,k}{\mathcal{A}}_2$ for $1\le j\le m_p$ and $1\le k\le m_{n+p}$ where $d\left(x_p\right)=m_p$ and $d\left(x_p^{-}\right)=m_{n+p}$. Let ${\mathcal{B}}_{j,k}$ be the formal set by letting $X_{p,j}=x_p$ and $Y_{p,k}=x_p^{-}$ and deleting elements of $U_{x_p,x_p^{-}}$ from $\mathcal{A}$ where $U_{x_p,x_p^{-}}=\dot{U}\backslash \{x_p,x_p^{-}\}$.

Because $S\in \mathcal{A}$ for each surface $S\in {\mathcal{B}}_{j,k}$,

$$\gamma \left(\mathcal{A}\right)\le \gamma \left({\mathcal{B}}_{j,k}\right).$$

(2)

Since for each surface $S_1\in \mathcal{B}$ there exists one and only one surface $S\in {\mathcal{B}}_{j,k}$ such that $S=x_p{x}_p^{-}{S}_1$, by Op1

$$S\sim S_1.$$

This concludes that

$$\gamma \left(\mathcal{B}\right)=\gamma \left({\mathcal{B}}_{j,k}\right).$$

(3)

By the definition of $\mathcal{A}$ and $\mathcal{B}$, S is formed from a surface $S_1\in \mathcal{B}$ by adding letters $x_p$ and $x_p^{-}$ for each $S\in \mathcal{A}$ and $S\notin {\mathcal{B}}_{j,k}$. This concludes that

$$\gamma \left(S_1\right)\le \gamma \left(S\right).$$

(4)

Hence the result is implied by applying (2)–(4). □

Now introduce an operation "≲" defined on $\mathsf{\Omega }$ with the following properties:

Op. set $\mathcal{F}=\mathcal{A}{X}_1\mathcal{B}{X}_2\mathcal{C}{X}_3\mathcal{D}{X}_4\mathcal{E}\in {\mathsf{\Omega }}_n$, where $\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}$, and $\mathcal{E}$ are formal sequences. If $X_1\in {\dot{U}}_{x_p^r},X_3\in {\dot{U}}_{x_p^{-r}},X_2\in {\dot{U}}_{x_q^{ϵ}},X_4\in {\dot{U}}_{x_q^{-ϵ}}$ with $r,ϵ\in \{+,-\}$, and $x_p\ne x_q$ for $1\le p\ne q\le n$, then

$$\mathcal{F}\lesssim {\mathcal{F}}_1{x}_p{x}_q{x}_p^{-}{x}_q^{-}$$

where $W={\dot{U}}_{x_p}\bigcup {\dot{U}}_{x_p^{-}}\bigcup {\dot{U}}_{x_q}\bigcup {\dot{U}}_{x_q^{-}}$ and ${\mathcal{F}}_1=\mathcal{A}\mathcal{D}\mathcal{C}\mathcal{B}\mathcal{E}\backslash W$; $[X_1,X_3]$ and $[X_2,X_4]$ are called poles of ${\mathcal{F}}_1$.

Lemma 2.

Let $\mathcal{F}\in {\mathsf{\Omega }}_n$ and let $\mathfrak{A}=\left\{\mathcal{A}\right|\mathcal{F}\lesssim \mathcal{A}{x}_p{x}_q{x}_p^{-}{x}_q^{-},1\le p\ne q\le n\}$. If $\mathcal{F}$ is non-planar, then

$$\gamma \left(\mathcal{F}\right)=\underset{\mathcal{A}\in \mathfrak{A}}{min}\left\{\gamma \left(\mathcal{A}\right)\right\}+1.$$

Proof. 

Because $\mathcal{F}$ is non-planar, there exists $1\le p\ne q\le n$ such that S has one of the following forms for each surface $S\in \mathcal{F}$:

$$A_1{x}_p{A}_2{x}_q^{-}{A}_3{x}_p^{-}{A}_4{x}_q{A}_5 A_1{x}_p^{-}{A}_2{x}_q^{-}{A}_3{x}_p{A}_4{x}_q{A}_5$$

$$A_1{x}_p^{-}{A}_2{x}_q{A}_3{x}_p{A}_4{x}_q^{-}{A}_5 A_1{x}_p{A}_2{x}_q{A}_3{x}_p^{-}{A}_4{x}_q^{-}{A}_5$$

where $A_i$ is a sequence of lowercase letters containing $x_j^{ϵ}$ for $1\le i\le 5$, $1\le j\le n$, $j\ne p$, $j\ne q$, and $ϵ\in \{+,-\}$. By Op2 and Op3,

$$\begin{array}{ccc}\hfill A_1{x}_p{A}_2{x}_q^{-}{A}_3{x}_p^{-}{A}_4{x}_q{A}_5& =& A_1{x}_p^{-}{A}_2{x}_q^{-}{A}_3{x}_p{A}_4{x}_q{A}_5\hfill \\ & =& A_1{x}_p^{-}{A}_2{x}_q{A}_3{x}_p{A}_4{x}_q^{-}{A}_5\hfill \\ & =& A_1{x}_p{A}_2{x}_q{A}_3{x}_p^{-}{A}_4{x}_q^{-}{A}_5\hfill \\ & \sim & A_1{A}_4{A}_3{A}_2{A}_5{x}_p{x}_q{x}_p^{-}{x}_q^{-}.\hfill \end{array}$$

Therefore,

$$\gamma \left(S\right)=\gamma \left(A_1{A}_4{A}_3{A}_2{A}_5\right)+1$$

where $A_1{A}_4{A}_3{A}_2{A}_5\in \mathcal{A}$ for some $\mathcal{A}\in \mathfrak{A}$.

For each surface $S_1\in \mathfrak{A}$, there exists $\mathcal{A}\in \mathfrak{A}$ such that $S_1\in \mathcal{A}$ and that $\mathcal{F}\lesssim \mathcal{A}{x}_p{x}_q{x}_p^{-}{x}_q^{-}$ with $1\le p,q\le n$ and $p\ne q$. This concludes that there is a surface $S\in \mathcal{F}$ such that $S\sim S_1{x}_p{x}_q{x}_p^{-}{x}_q^{-}$.

Hence the result is clear. □

It is obvious that the following result holds by Lemma 2.

Theorem 3.

Let $\mathcal{F}$ be a non-planar formal set. Then $\gamma \left(\mathcal{F}\right)=g$ if and only if there exists a positive integer g such that ${\mathcal{F}}_g$ is planar and ${\mathcal{F}}_k$ is non-planar for $k\le g-1$, where ${\mathcal{F}}_i$ is obtained by applying Op i times from $\mathcal{F}$ for $i\le g$.

Suppose that G is a cubic graph with a Hamilton cycle and that G has $n+1$ cotree edges with $n\ge 1$. Then G has $2n$ vertices. Without loss of generality, set $C=v_1{v}_2\cdots v_{2n}{v}_1$ to be a Hamilton cycle of G and set $x=v_1{v}_{2n}$. Choose $T=v_1{v}_2\cdots v_{2n}$ as a spanning tree of G and denote other cotree edges with $x_1,x_2,\cdots$$,x_n$. A joint tree $\tilde{T}$ is obtained by splitting cotree edges a into two semi-edges with labels a and $a^{-}$ for $a\in \{x,x_i|1\le i\le n\}$. Because each vertex has two rotations, set ${\sigma }_1\left(v_1\right)=(x,x_{l_1},v_1{v}_2)$, ${\sigma }_2\left(v_1\right)=(x,v_1{v}_2,x_{l_1})$, ${\sigma }_1\left(v_{2n}\right)=(v_{2n-1}{v}_{2n},x_{l_{2n}},x)$, ${\sigma }_2\left(v_{2n}\right)=(v_{2n-1}{v}_{2n},x,x_{l_{2n}})$, ${\sigma }_1\left(v_j\right)=(v_{j-1}{v}_j,x_{l_j},v_j{v}_{j+1})$, ${\sigma }_2\left(v_j\right)=(v_{j-1}{v}_j,v_j{v}_{j+1},x_{l_j})$ for $2\le j\le 2n-1$, $1\le l_j\le n$. Set the formal set $\mathcal{F}=x{Z}_1{Z}_2\cdots Z_{2n}{x}^{-}{\overline{Z}}_{2n}\cdots {\overline{Z}}_2{\overline{Z}}_1$. Here, if $x_k$ are adjacent to $v_i$ and $v_j$ with $i<j$, then $Z_i=X_k$ and $Z_j=Y_k$, where $X_k\in \{x_k,\varnothing \}$, $Y_k\in \{x_k^{-},\varnothing \}$ for $1\le k\le n$, $1\le i,j\le 2n$. Let each vertex have a clockwise rotation in any joint tree. There is a bijection between $\mathcal{F}$ and the set of associate surfaces of all joint trees of G. The bijection is obtained by letting $Z_j\ne \varnothing$ if and only if $\sigma \left(v_j\right)={\sigma }_1\left(v_j\right)$, ${\overline{Z}}_j\ne \varnothing$ if and only if $\sigma \left(v_j\right)={\sigma }_2\left(v_j\right)$ for $1\le j\le 2n$. Therefore, we have the following result.

Theorem 4.

Let G be a Hamiltonian cubic graph, then its set of all associate surfaces is a cubic basic set.

Now we show a simple example to determine the genus of a cubic graph with a Hamilton cycle.

Example 2.

Let H be the cubic graph in Figure 2.

Obviously, H is non-planar. Choose the Hamilton path $u_1{u}_2\cdots u_{18}$ as the spanning tree T. Then the set of all associate surfaces of H is the formal set

$$\begin{array}{ccc}\hfill \mathcal{F}& =& x{X}_1{X}_2{X}_3{X}_4{X}_5{X}_6{X}_7{Y}_3{X}_8{Y}_5{X}_9{Y}_7{Y}_1{Y}_2{Y}_8{Y}_4{Y}_9{Y}_6{x}^{-}\hfill \\ & & {\overline{Y}}_6{\overline{Y}}_9{\overline{Y}}_4{\overline{Y}}_8{\overline{Y}}_2{\overline{Y}}_1{\overline{Y}}_7{\overline{X}}_9{\overline{Y}}_5{\overline{X}}_8{\overline{Y}}_3{\overline{X}}_7{\overline{X}}_6{\overline{X}}_5{\overline{X}}_4{\overline{X}}_3{\overline{X}}_2{\overline{X}}_1\hfill \end{array}$$

Let $[X_1,{\overline{Y}}_1]$ and $[{\overline{Y}}_4,{\overline{X}}_4]$ be poles. By applying Op

$$\begin{array}{ccc}\hfill \mathcal{F}& \lesssim & {\mathcal{F}}_1{x}_1{x}_4{x}_1^{-}{x}_4^{-}\hfill \end{array}$$

where ${\mathcal{F}}_1=x{\overline{Y}}_7{\overline{X}}_9{\overline{Y}}_5{\overline{X}}_8{\overline{Y}}_3{\overline{X}}_7{\overline{X}}_6{\overline{X}}_5{\overline{Y}}_8{\overline{Y}}_2{X}_2{X}_3{X}_5{X}_6{X}_7{Y}_3{X}_8{Y}_5{X}_9{Y}_7{Y}_2{Y}_8{Y}_9{Y}_6{x}^{-}{\overline{Y}}_6{\overline{Y}}_9{\overline{X}}_3{\overline{X}}_2$.

Because ${\mathcal{A}}_1={\overline{Y}}_7{\overline{X}}_9{\overline{Y}}_5{\overline{X}}_8{\overline{Y}}_3{\overline{X}}_7{\overline{X}}_6{\overline{X}}_5{\overline{Y}}_8{\overline{Y}}_2{X}_2{X}_3{X}_5{X}_6{X}_7{Y}_3{X}_8{Y}_5{X}_9{Y}_7{Y}_2{Y}_8{Y}_9{Y}_6$ contains an eliminable letter $x_2$, by Lemma 1

$$\gamma \left({\mathcal{A}}_1\right)=\gamma \left(\mathcal{A}\right)$$

where

$$\mathcal{A}={\overline{Y}}_7{\overline{X}}_9{\overline{Y}}_5{\overline{X}}_8{\overline{Y}}_3{\overline{X}}_7{\overline{X}}_6{\overline{X}}_5{\overline{Y}}_8{X}_3{X}_5{X}_6{X}_7{Y}_3{X}_8{Y}_5{X}_9{Y}_7{Y}_8{Y}_9{Y}_6.$$

Because $Acc\left(\mathcal{A}\right)$ contains an independent set $I=\{u_{2,1,2},u_{3,1,1},u_{5,2,1},u_{6,2,1},u_{7,2,2},u_{8,1,2},u_{9,1,1}\}$ where $u_{3,1,1}=[X_3,Y_3]$, $u_{5,1,1}=[{\overline{X}}_5,Y_5]$, $u_{6,1,1}=[{\overline{X}}_6,Y_6]$, $u_{7,1,2}=[{\overline{Y}}_7,{\overline{X}}_7]$, $u_{8,1,2}=[{\overline{Y}}_8,X_8]$, $u_{9,1,1}=[X_9,Y_9],$ it implies that ${\mathcal{F}}_1$ is planar. By applying Theorem 3

$$\gamma \left(\mathcal{F}\right)=1.$$

Therefore

$$\gamma \left(H\right)=1.$$

Let each vertex have a clockwise rotation in $\tilde{T}$ in Figure 2. Then the genus of the associate surface of $\tilde{T}$ is equal to 1.

In the rest of this section, several problems are given for further study.

Problem 1.

Characterize these graphs of genus $g\ge 1$.

There exist several classical criteria to characterize planar graphs. If a graph contain no K3,3 as a minor, then Gagarin et al. obtained the complete list of obstructions for the torus [52]. How to generalize these results for a graph of genus $g\ge 1$?

Problem 2.

Is there a polynomial algorithm to determine the genus of a cubic graph with a given Hamilton cycle?

Thomassen proved that determining the genus of a cubic graph is NP-complete [53]. Its set of all associate surfaces is a cubic basic formal set for a cubic graph with a given Hamilton cycle. Gavril provided an algorithm to find the maximum independent set of an overlap graph [54]. Because the accompanying graph of a formal set is an overlap graph, there is an algorithm to determine whether a formal set is planar in a polynomial time. Is there an algorithm to determine the genus of a cubic basic formal set in a polynomial time?

Problem 3.

Find a new algebraic or combinatorial construction to study the genus of a graph, especially 3-connected cubic graphs.

Conjecture 3.

For a complete tripartite graph $K_{m,n,l}$ with$m\ge n\ge l\ge 1$

$$\gamma \left(K_{m,n,l}\right)=⌈\frac{(m-2)(n+l-2)}{4}⌉.$$

White proposed the conjecture above [16]. Although many cases have been obtained, which are listed in Appendix A, this conjecture is still open.