Euler-genus distributions of cubic Halin graphs

Jinlian Zhang; Xuhui Peng

arXiv:1903.07060·math.CO·February 20, 2020

Euler-genus distributions of cubic Halin graphs

Jinlian Zhang, Xuhui Peng

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TL;DR

This paper develops a recurrence relation and explicit formulas for the Euler-genus distribution of cubic caterpillar-Halin graphs, advancing understanding of their topological embeddings.

Contribution

It introduces a recurrence relation for the Euler-genus polynomial of cubic caterpillar-Halin graphs and provides explicit formulas for embeddings into surfaces with low Euler-genus.

Findings

01

Recurrence relation for Euler-genus polynomial derived

02

Explicit formulas for embeddings into surfaces with Euler-genus 0, 1, 2 obtained

03

Advances in calculating topological embeddings of cubic Halin graphs

Abstract

Gross derived an $O (n^{2})$ -time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of overlap matrix, we get a recurrence relation for the Euler-genus polynomial of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of cubic caterpillar-Halin graph into a surface with Euler-genus 0, 1 and 2 are also obtained.

Equations191

γ^{E} = {2 j, if S = O_{j}, k, if S = N_{k} .

γ^{E} = {2 j, if S = O_{j}, k, if S = N_{k} .

e_{1}, \dots e_{k}, e_{1, 1}, \dots, e_{1, m_{1}}, e_{2, 1}, \dots, e_{2, m_{2}}, \dots, e_{k - 1, 1}, \dots, e_{k - 1, m_{k - 1}} e_{k, 1}, \dots, e_{k, m_{k}} .

e_{1}, \dots e_{k}, e_{1, 1}, \dots, e_{1, m_{1}}, e_{2, 1}, \dots, e_{2, m_{2}}, \dots, e_{k - 1, 1}, \dots, e_{k - 1, m_{k - 1}} e_{k, 1}, \dots, e_{k, m_{k}} .

m_{ij} = ⎩ ⎨ ⎧ 1, 1, 0, if i = j and e_{i} is twisted; if i \neq = j and the restriction of the underlying pure rotation system to the subgraph T + e_{i} + e_{j} is nonplanar; otherwise.

m_{ij} = ⎩ ⎨ ⎧ 1, 1, 0, if i = j and e_{i} is twisted; if i \neq = j and the restriction of the underlying pure rotation system to the subgraph T + e_{i} + e_{j} is nonplanar; otherwise.

E^{1} (t_{1}, t_{2}, \dots, t_{k}, z) = m_{1}, m_{2}, \dots, m_{k} ⩾ 1 \sum E_{m_{1}, m_{2}, \dots, m_{k}} (z) t_{1}^{m_{1}} t_{2}^{m_{2}} \dots t_{k}^{m_{k}} .

E^{1} (t_{1}, t_{2}, \dots, t_{k}, z) = m_{1}, m_{2}, \dots, m_{k} ⩾ 1 \sum E_{m_{1}, m_{2}, \dots, m_{k}} (z) t_{1}^{m_{1}} t_{2}^{m_{2}} \dots t_{k}^{m_{k}} .

a_{1} b_{1} 0 b_{1} a_{2} b_{2} b_{2} a_{3} ⋱ 0 ⋱ ⋱ b_{m - 1} b_{m - 1} a_{m} .

a_{1} b_{1} 0 b_{1} a_{2} b_{2} b_{2} a_{3} ⋱ 0 ⋱ ⋱ b_{m - 1} b_{m - 1} a_{m} .

x_{0} y_{0} x_{i} y_{i} z_{1} z_{i} z_{k} = (x_{1}, \dots, x_{k}) \in (Z_{2})^{k}, = (y_{1}, \dots, y_{k - 1}) \in (Z_{2})^{k - 1}, = (x_{i, 1}, \dots, x_{i, m_{i}}) \in (Z_{2})^{m_{i}}, i = 1, \dots, k, = (y_{i, 1}, \dots, y_{i, m_{i} - 1}) \in (Z_{2})^{m_{i} - 1}, i = 1, \dots, k, = (z_{1, 1} \dots, z_{1, m_{1} + 1}) \in (Z_{2})^{m_{1} + 1}, = (z_{i, 0}, z_{i, 1}, \dots, z_{i, m_{i} + 1}) \in (Z_{2})^{m_{i} + 2}, i = 2, \dots, k - 1, = (z_{k, 0}, z_{k, 1}, \dots, z_{k - 1, m_{k}}) \in (Z_{2})^{m_{k} + 1},

x_{0} y_{0} x_{i} y_{i} z_{1} z_{i} z_{k} = (x_{1}, \dots, x_{k}) \in (Z_{2})^{k}, = (y_{1}, \dots, y_{k - 1}) \in (Z_{2})^{k - 1}, = (x_{i, 1}, \dots, x_{i, m_{i}}) \in (Z_{2})^{m_{i}}, i = 1, \dots, k, = (y_{i, 1}, \dots, y_{i, m_{i} - 1}) \in (Z_{2})^{m_{i} - 1}, i = 1, \dots, k, = (z_{1, 1} \dots, z_{1, m_{1} + 1}) \in (Z_{2})^{m_{1} + 1}, = (z_{i, 0}, z_{i, 1}, \dots, z_{i, m_{i} + 1}) \in (Z_{2})^{m_{i} + 2}, i = 2, \dots, k - 1, = (z_{k, 0}, z_{k, 1}, \dots, z_{k - 1, m_{k}}) \in (Z_{2})^{m_{k} + 1},

Δ_{m_{1}, \dots, m_{k}}^{X, Y, Z} = L_{k}^{x_{0}, y_{0}} M_{1} M_{2} ⋮ M_{k - 1} M_{k} M_{1}^{T} L_{m_{1}}^{x_{1}, y_{1}} M_{2}^{T} L_{m_{2}}^{x_{2}, y_{2}} 0 \dots ⋱ M_{k - 1}^{T} 0 L_{m_{k}}^{x_{k - 1}, y_{k - 1}} M_{k}^{T} L_{m_{k}}^{x_{k}, y_{k}},

Δ_{m_{1}, \dots, m_{k}}^{X, Y, Z} = L_{k}^{x_{0}, y_{0}} M_{1} M_{2} ⋮ M_{k - 1} M_{k} M_{1}^{T} L_{m_{1}}^{x_{1}, y_{1}} M_{2}^{T} L_{m_{2}}^{x_{2}, y_{2}} 0 \dots ⋱ M_{k - 1}^{T} 0 L_{m_{k}}^{x_{k - 1}, y_{k - 1}} M_{k}^{T} L_{m_{k}}^{x_{k}, y_{k}},

z_{1, 1} z_{1, 2} ⋮ z_{1, m_{1} - 1} z_{1, m_{1}} 00 ⋮ 0 z_{2, 0} 00 ⋮ 00 \dots \dots ⋮ \dots \dots 00 ⋮ 00 .

z_{1, 1} z_{1, 2} ⋮ z_{1, m_{1} - 1} z_{1, m_{1}} 00 ⋮ 0 z_{2, 0} 00 ⋮ 00 \dots \dots ⋮ \dots \dots 00 ⋮ 00 .

z_{i - 1, m_{i - 1} + 1} 0 ⋮ 00, z_{i, 1} z_{i, 2} ⋮ z_{i, m_{i} - 1} z_{i, m_{i}}, 00 ⋮ 0 z_{i + 1, 0};

z_{i - 1, m_{i - 1} + 1} 0 ⋮ 00, z_{i, 1} z_{i, 2} ⋮ z_{i, m_{i} - 1} z_{i, m_{i}}, 00 ⋮ 0 z_{i + 1, 0};

00 ⋮ 00 \dots \dots ⋮ \dots \dots 00 ⋮ 00 z_{k - 1, m_{k - 1} + 1} 0 ⋮ 00 z_{k, 1} z_{k, 2} z_{k, m_{k} - 1} z_{k, m_{k}} .

00 ⋮ 00 \dots \dots ⋮ \dots \dots 00 ⋮ 00 z_{k - 1, m_{k - 1} + 1} 0 ⋮ 00 z_{k, 1} z_{k, 2} z_{k, m_{k} - 1} z_{k, m_{k}} .

00 L_{k - 1}^{x_{0}, y_{0}} \dots 0 y_{k - 1} 00 ⋮ 0 y_{k - 1} x_{k} M_{1} ⋮ M_{k - 1} 00 ⋮ 000 00 ⋮ 000 \dots \dots ⋮ \dots \dots \dots 00 ⋮ 000 z_{k - 1, m_{k - 1} + 1} 0 ⋮ 000 z_{k, 1} z_{k, 2} ⋮ z_{k, m_{k} - 2} z_{k, m_{k} - 1} z_{k, m_{k}} M_{1}^{T} L_{m_{1}}^{x_{1}, y_{1}} \dots ⋱ 0 M_{k - 1}^{T} L_{m_{k - 1}}^{x_{k - 1}, y_{k - 1}} 00 ⋮ 0 z_{k - 1, m_{k - 1} + 1} z_{k, 1} 00 ⋮ 00 z_{k, 2} \dots \dots ⋮ \dots \dots \dots 00 ⋮ 00 z_{k, m_{k} - 2} 00 ⋮ 00 z_{k, m_{k} - 1} 00 ⋮ 00 z_{k, m_{k}} 0 x_{k, 1} y_{k, 1} y_{k, 1} ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ x_{k, m_{k} - 2} y_{k, m_{k} - 2} y_{k, m_{k} - 2} x_{k, m_{k} - 1} y_{k, m_{k} - 1} y_{k, m_{k} - 1} x_{k, m_{k}}

00 L_{k - 1}^{x_{0}, y_{0}} \dots 0 y_{k - 1} 00 ⋮ 0 y_{k - 1} x_{k} M_{1} ⋮ M_{k - 1} 00 ⋮ 000 00 ⋮ 000 \dots \dots ⋮ \dots \dots \dots 00 ⋮ 000 z_{k - 1, m_{k - 1} + 1} 0 ⋮ 000 z_{k, 1} z_{k, 2} ⋮ z_{k, m_{k} - 2} z_{k, m_{k} - 1} z_{k, m_{k}} M_{1}^{T} L_{m_{1}}^{x_{1}, y_{1}} \dots ⋱ 0 M_{k - 1}^{T} L_{m_{k - 1}}^{x_{k - 1}, y_{k - 1}} 00 ⋮ 0 z_{k - 1, m_{k - 1} + 1} z_{k, 1} 00 ⋮ 00 z_{k, 2} \dots \dots ⋮ \dots \dots \dots 00 ⋮ 00 z_{k, m_{k} - 2} 00 ⋮ 00 z_{k, m_{k} - 1} 00 ⋮ 00 z_{k, m_{k}} 0 x_{k, 1} y_{k, 1} y_{k, 1} ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ x_{k, m_{k} - 2} y_{k, m_{k} - 2} y_{k, m_{k} - 2} x_{k, m_{k} - 1} y_{k, m_{k} - 1} y_{k, m_{k} - 1} x_{k, m_{k}}

E_{m_{1}, \dots, m_{k}} (z) = 2 δ_{m_{1}, \dots, m_{k}} (z),

E_{m_{1}, \dots, m_{k}} (z) = 2 δ_{m_{1}, \dots, m_{k}} (z),

\scriptsize{\left[\begin{array}[]{ccccc}\begin{array}[]{ccccc}&&&&0\\ &\normalsize{\text{$\mathcal{L}_{k-1}^{\textbf{x}_{0},\textbf{y}_{0}}$}}&&&\vdots\\ &&&&0\\ &&&&0\\ 0&\cdots&0&\text{~{}~{}~{}~{}~{}~{}~{}0~{}~{}~{}~{}~{}~{}~{}}&0\end{array}&\normalsize{\text{$M^{T}_{1}$}}&\cdots&\normalsize{\text{$M^{T}_{k-1}$}}&\begin{array}[]{ccccc}0&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&0&0\\ z_{k-1,m_{k-1}+1}&0&\cdots&0&0\\ 0&~{}~{}0{}{}&\cdots&~{}~{}~{}~{}0{}{}{}{}&~{}~{}~{}~{}0{}{}{}{}\end{array}\\ \normalsize{\text{$M_{1}$}}&\normalsize{\text{$\mathcal{L}$}}_{m_{1}}^{\textbf{x}_{1},\textbf{y}_{1}}&&\\ \vdots&&\ddots&&\normalsize{\textbf{0}}\\ \normalsize{\text{$M_{k-1}$}}&&&\normalsize{\text{$\mathcal{L}$}}_{m_{k-1}}^{\textbf{x}_{k-1},\textbf{y}_{k-1}}&\\ \begin{array}[]{ccccc}0&~{}~{}~{}\cdots{}{}{}&0&\tiny{z_{k-1,m_{k-1}+1}}&0\\ 0&\cdots&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&0&0&0\\ 0&\cdots&0&0&0\end{array}&&\normalsize{\textbf{0}}&&\begin{array}[]{ccccc}\text{~{}~{}~{}~{}~{}$x_{k,1}$~{}~{}~{}~{}}&y_{k,1}&&&\\ \text{~{}~{}$y_{k,1}$~{}}&\ddots{}&~{}\ddots&&\\ &\ddots{}{}&\ddots&\ddots\\ &&\ddots{}{}&x_{k,{m_{k}-1}}&y_{k,m_{k}-1}\\ &&&y_{k,{m_{k}-1}}&x_{k,m_{k}}\end{array}\end{array}\right]}

\scriptsize{\left[\begin{array}[]{ccccc}\begin{array}[]{ccccc}&&&&0\\ &\normalsize{\text{$\mathcal{L}_{k-1}^{\textbf{x}_{0},\textbf{y}_{0}}$}}&&&\vdots\\ &&&&0\\ &&&&0\\ 0&\cdots&0&\text{~{}~{}~{}~{}~{}~{}~{}0~{}~{}~{}~{}~{}~{}~{}}&0\end{array}&\normalsize{\text{$M^{T}_{1}$}}&\cdots&\normalsize{\text{$M^{T}_{k-1}$}}&\begin{array}[]{ccccc}0&0&\cdots&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&0&0\\ z_{k-1,m_{k-1}+1}&0&\cdots&0&0\\ 0&~{}~{}0{}{}&\cdots&~{}~{}~{}~{}0{}{}{}{}&~{}~{}~{}~{}0{}{}{}{}\end{array}\\ \normalsize{\text{$M_{1}$}}&\normalsize{\text{$\mathcal{L}$}}_{m_{1}}^{\textbf{x}_{1},\textbf{y}_{1}}&&\\ \vdots&&\ddots&&\normalsize{\textbf{0}}\\ \normalsize{\text{$M_{k-1}$}}&&&\normalsize{\text{$\mathcal{L}$}}_{m_{k-1}}^{\textbf{x}_{k-1},\textbf{y}_{k-1}}&\\ \begin{array}[]{ccccc}0&~{}~{}~{}\cdots{}{}{}&0&\tiny{z_{k-1,m_{k-1}+1}}&0\\ 0&\cdots&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&\cdots&0&0&0\\ 0&\cdots&0&0&0\end{array}&&\normalsize{\textbf{0}}&&\begin{array}[]{ccccc}\text{~{}~{}~{}~{}~{}$x_{k,1}$~{}~{}~{}~{}}&y_{k,1}&&&\\ \text{~{}~{}$y_{k,1}$~{}}&\ddots{}&~{}\ddots&&\\ &\ddots{}{}&\ddots&\ddots\\ &&\ddots{}{}&x_{k,{m_{k}-1}}&y_{k,m_{k}-1}\\ &&&y_{k,{m_{k}-1}}&x_{k,m_{k}}\end{array}\end{array}\right]}

λ_{m_{1}, m_{2}, \dots, m_{k}} (z) = j = 0 \sum k + m_{1} + \dots + m_{k} λ_{m_{1}, m_{2}, \dots, m_{k}} [j] z^{j},

λ_{m_{1}, m_{2}, \dots, m_{k}} (z) = j = 0 \sum k + m_{1} + \dots + m_{k} λ_{m_{1}, m_{2}, \dots, m_{k}} [j] z^{j},

Φ_{m + 1}^{x, y, z} = x_{0} z_{1} z_{2} z_{3} ⋮ z_{m - 1} z_{m} z_{1} x_{1} y_{1} z_{2} y_{1} x_{2} y_{2} 0 z_{3} y_{2} x_{3} ⋱ \dots ⋱ ⋱ y_{m - 2} z_{n - 1} 0 y_{m - 2} x_{m - 1} y_{m - 1} z_{n} y_{m - 1} x_{m} .

Φ_{m + 1}^{x, y, z} = x_{0} z_{1} z_{2} z_{3} ⋮ z_{m - 1} z_{m} z_{1} x_{1} y_{1} z_{2} y_{1} x_{2} y_{2} 0 z_{3} y_{2} x_{3} ⋱ \dots ⋱ ⋱ y_{m - 2} z_{n - 1} 0 y_{m - 2} x_{m - 1} y_{m - 1} z_{n} y_{m - 1} x_{m} .

Φ_{m + 1} = {Φ_{m + 1}^{x, y, z} : x \in Z_{2}^{m + 1}, y \in Z_{2}^{m - 1}, z \in Z_{2}^{m}}

Φ_{m + 1} = {Φ_{m + 1}^{x, y, z} : x \in Z_{2}^{m + 1}, y \in Z_{2}^{m - 1}, z \in Z_{2}^{m}}

δ_{m_{1}, 0} (z) = ϕ_{m_{1} + 2} (z) and δ_{m_{1}, 1} (z) = ϕ_{m_{1} + 3} (z) .

δ_{m_{1}, 0} (z) = ϕ_{m_{1} + 2} (z) and δ_{m_{1}, 1} (z) = ϕ_{m_{1} + 3} (z) .

L_{m} (z) = (1 + 2 z) L_{m - 1} (z) + 4 z^{2} L_{m - 2} (z)

L_{m} (z) = (1 + 2 z) L_{m - 1} (z) + 4 z^{2} L_{m - 2} (z)

ϕ_{m + 1} (z) = (1 + 4 z) ϕ_{m} (z) + 16 z^{2} ϕ_{m - 1} (z) + 2^{m} z^{2} L_{m - 1} (z)

ϕ_{m + 1} (z) = (1 + 4 z) ϕ_{m} (z) + 16 z^{2} ϕ_{m - 1} (z) + 2^{m} z^{2} L_{m - 1} (z)

\displaystyle\phi(t,z):=\sum_{m\geqslant 2}\phi_{m}(z)t^{m}=\frac{t^{2}\big{(}1+3z+4z^{2}-2(1+5z+4z^{2}+2z^{3})t-16z^{2}(2+6z+5z^{2})t^{2}-168z^{4}(1+z)t^{3}\big{)}}{(1-2t-4tz-16z^{2}t^{2})(1-t-4tz-16z^{2}t^{2})}.

\displaystyle\phi(t,z):=\sum_{m\geqslant 2}\phi_{m}(z)t^{m}=\frac{t^{2}\big{(}1+3z+4z^{2}-2(1+5z+4z^{2}+2z^{3})t-16z^{2}(2+6z+5z^{2})t^{2}-168z^{4}(1+z)t^{3}\big{)}}{(1-2t-4tz-16z^{2}t^{2})(1-t-4tz-16z^{2}t^{2})}.

\left(\begin{array}[]{ccccccccccc}x_{1}&y_{1}&z_{1,1}&z_{1,2}&\cdots&z_{1,m_{1}-1}&z_{1,m_{1}}&z_{1,m_{1}+1}&0&\cdots&0\\ y_{1}&x_{2}&0&0&\cdots&0&z_{2,0}&z_{2,1}&z_{2,2}&\cdots&z_{2,m_{2}}\\ z_{1,1}&0&x_{1,1}&y_{1,1}\\ z_{1,2}&0&y_{1,1}&x_{1,2}&\ddots\\ \vdots&\vdots&&\ddots&\ddots&y_{1,m_{1}-2}&&&\normalsize{\textbf{0}}&\\ z_{1,m_{1}-1}&0&&&y_{1,m_{1}-2}&x_{1,m_{1}-1}&y_{1,m_{1}-1}\\ z_{1,m_{1}}&z_{2,0}&&&&y_{1,m_{1}-1}&x_{1,m_{1}}\\ z_{1,m_{1}+1}&z_{2,1}&&&&&&x_{2,1}&y_{2,1}\\ 0&z_{2,2}&&&&&&y_{2,1}&x_{2,2}&\ddots\\ \vdots&\vdots&&&\normalsize{\textbf{0}}&&&&\ddots&\ddots&y_{2,m_{2}-1}\\ 0&z_{2,m_{2}}&&&&&&&&y_{2,m_{2}-1}&x_{2,m_{2}}\end{array}\right)

\left(\begin{array}[]{ccccccccccc}x_{1}&y_{1}&z_{1,1}&z_{1,2}&\cdots&z_{1,m_{1}-1}&z_{1,m_{1}}&z_{1,m_{1}+1}&0&\cdots&0\\ y_{1}&x_{2}&0&0&\cdots&0&z_{2,0}&z_{2,1}&z_{2,2}&\cdots&z_{2,m_{2}}\\ z_{1,1}&0&x_{1,1}&y_{1,1}\\ z_{1,2}&0&y_{1,1}&x_{1,2}&\ddots\\ \vdots&\vdots&&\ddots&\ddots&y_{1,m_{1}-2}&&&\normalsize{\textbf{0}}&\\ z_{1,m_{1}-1}&0&&&y_{1,m_{1}-2}&x_{1,m_{1}-1}&y_{1,m_{1}-1}\\ z_{1,m_{1}}&z_{2,0}&&&&y_{1,m_{1}-1}&x_{1,m_{1}}\\ z_{1,m_{1}+1}&z_{2,1}&&&&&&x_{2,1}&y_{2,1}\\ 0&z_{2,2}&&&&&&y_{2,1}&x_{2,2}&\ddots\\ \vdots&\vdots&&&\normalsize{\textbf{0}}&&&&\ddots&\ddots&y_{2,m_{2}-1}\\ 0&z_{2,m_{2}}&&&&&&&&y_{2,m_{2}-1}&x_{2,m_{2}}\end{array}\right)

x_{1} 0 z_{1, 1} z_{1, 2} ⋮ z_{1, m_{1}} z_{1, m_{1} + 1} 0 ⋮ 0 0000 ⋮ 000 ⋮ 0 z_{1, 1} 0 x_{1, 1} y_{1, 1} z_{1, 2} 0 y_{1, 1} x_{1, 2} ⋱ \dots \dots ⋱ ⋱ y_{1, m_{1} - 1} 0 z_{1, m_{1}} 0 y_{1, m_{1} - 1} x_{1, m_{1}} z_{1, m_{1} + 1} 0 x_{2, 1} y_{2, 1} 00 y_{2, 1} x_{2, 2} ⋱ \dots \dots 0 ⋱ ⋱ y_{2, m_{2} - 1} 00 y_{2, m_{2} - 1} x_{2, m_{2}}

x_{1} 0 z_{1, 1} z_{1, 2} ⋮ z_{1, m_{1}} z_{1, m_{1} + 1} 0 ⋮ 0 0000 ⋮ 000 ⋮ 0 z_{1, 1} 0 x_{1, 1} y_{1, 1} z_{1, 2} 0 y_{1, 1} x_{1, 2} ⋱ \dots \dots ⋱ ⋱ y_{1, m_{1} - 1} 0 z_{1, m_{1}} 0 y_{1, m_{1} - 1} x_{1, m_{1}} z_{1, m_{1} + 1} 0 x_{2, 1} y_{2, 1} 00 y_{2, 1} x_{2, 2} ⋱ \dots \dots 0 ⋱ ⋱ y_{2, m_{2} - 1} 00 y_{2, m_{2} - 1} x_{2, m_{2}}

λ_{m_{1}, m_{2}} (z)

λ_{m_{1}, m_{2}} (z)

{λ_{m_{1}, 1} (z) λ_{m_{1}, 2} (z) = (1 + 2 z) ϕ_{m_{1} + 1} (z) + 2^{m_{1} + 1} z^{2} L_{m_{1}} (z), = (1 + 2 z) λ_{m_{1}, 1} (z) + 4 z^{2} ϕ_{m_{1} + 1} (z) .

{λ_{m_{1}, 1} (z) λ_{m_{1}, 2} (z) = (1 + 2 z) ϕ_{m_{1} + 1} (z) + 2^{m_{1} + 1} z^{2} L_{m_{1}} (z), = (1 + 2 z) λ_{m_{1}, 1} (z) + 4 z^{2} ϕ_{m_{1} + 1} (z) .

E_{m_{1}, m_{2}} (z)

E_{m_{1}, m_{2}} (z)

δ_{m_{1}, m_{2}} (z)

δ_{m_{1}, m_{2}} (z)

λ_{m_{1}, m_{2}, \dots, m_{k - 1}, m_{k}} (z) = (1 + 2 z) λ_{m_{1}, m_{2}, \dots, m_{k - 1}, m_{k} - 1} (z) + 4 z^{2} λ_{m_{1}, m_{2}, \dots, m_{k - 1}, m_{k} - 2} (z)

λ_{m_{1}, m_{2}, \dots, m_{k - 1}, m_{k}} (z) = (1 + 2 z) λ_{m_{1}, m_{2}, \dots, m_{k - 1}, m_{k} - 1} (z) + 4 z^{2} λ_{m_{1}, m_{2}, \dots, m_{k - 1}, m_{k} - 2} (z)

λ_{m_{1}, \dots, m_{k - 1}, 0} (z) = \frac{1}{2} E_{m_{1}, \dots, m_{k - 1}} (z),

λ_{m_{1}, \dots, m_{k - 1}, 0} (z) = \frac{1}{2} E_{m_{1}, \dots, m_{k - 1}} (z),

λ_{m_{1}, \dots, m_{k - 1}, 1} (z) = \frac{1 + 2 z}{2} E_{m_{1}, \dots, m_{k - 1}} (z) + 2^{m_{k - 1} + 3} z^{2} \cdot λ_{m_{1}, \dots, m_{k - 1}} (z) .

λ_{m_{1}, m_{2}, \dots, m_{k - 1}, 1} (z)

λ_{m_{1}, m_{2}, \dots, m_{k - 1}, 1} (z)

λ_{m_{1}, m_{2}, \dots, m_{k - 1}, 2} (z)

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TopicsAdvanced Graph Theory Research · Graph theory and applications · Limits and Structures in Graph Theory

Full text

Euler-genus distributions of cubic caterpillar-Halin graphs

Jinlian Zhang*†*

*department of Mathematics, Hunan University, 410082 Changsha, China

* Xuhui Peng♠

*department of statistics, Hunan Normal University, 410081 Changsha, China

Abstract

Gross [6] derived an $O(n^{2})$ -time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of overlap matrix, we get a recurrence relation for the Euler-genus polynomial of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of cubic caterpillar-Halin graph into a surface with Euler-genus 0, 1 and 2 are also obtained.

Keywords: Embedding distributions; Euler-genus polynomial; Cubic Halin graph

MSC 2000: Primary:05C10; Secondary:05A15; 05C30.

††footnotetext: † Email: [email protected].; ♠ Email: [email protected]

1 Introduction

There are two kinds of surfaces in topology. One is the orientable surfaces $O_{j}$ , which has $j$ handles $(j\geq 0)$ and the other is the non-orientable surfaces $N_{k}$ , which has $k$ crosscaps $(k>0)$ . The research of graph embeddings on non-orientable surfaces was began by Chen, Gross and Rieper [1]. The authors in [4] introduced a combinatorial way to calculate the Euler-genus polynomial of a graph. Recently, an $O(n^{2})$ -time algorithm to calculate the genus distribution of any cubic Halin graph $H_{T}$ was derived by Gross [6]. For linear graph families, Stahl [11] calculated their genus distributions. Enami [7] made researches on the inequivalent embeddings for 3-connected 3-regular planar graph. However, there still no results on recursive formulas for Euler-genus distributions of any cubic Halin graph. In this paper, we make researches on cubic caterpillar-Halin graphs and obtain their recurrence relation for Euler-genus polynomial.

1.1 Euler-genus polynomial

The Euler-genus $\gamma^{E}$ of a surface $S$ is

[TABLE]

In this paper, $S_{i}$ denotes a surface $S$ with Euler-genus $i$ .

A pure rotation system $\rho$ of graph $G=(V(G),E(G))$ is the set $\{\rho_{v}:v\in V(G)\},$ where $\rho_{v}$ is a cyclic permutation of edges incident to $v$ . Assume $\lambda$ is a mapping $E(G)\rightarrow\{0,1\}$ . We say edge $e$ is untwisted if $\lambda(e)=0$ , and we say it is twisted if $\lambda(e)=1$ . This pair $(\rho,\lambda)$ is called a general rotation system for graph $G$ . Let $T$ be a spanning tree of graph $G$ . If $\lambda(e)=0,\forall e\in E(T),$ this general rotation system $(\rho,\lambda)$ is called a $T$ -rotation system. For any two embeddings of $G$ , if their $T$ -rotation systems are combinatorially equivalent, we consider them to be equivalent. Let $\varepsilon_{G}(i)$ be the number of equivalent embeddings of $G$ into the surface $S_{i},$ for $i\geqslant 0.$ The Euler-genus distribution of graph $G$ is the sequence $\varepsilon_{G}(0),\varepsilon_{G}(1),\cdots,\varepsilon_{G}(i),\cdots$ and the Euler-genus polynomial of graph $G$ is $\mathcal{E}_{G}(z)=\displaystyle\sum_{i=0}^{\infty}\varepsilon_{G}(i)z^{i}.$

1.2 Cubic caterpillar-Halin graphs

A cubic Halin graph $H_{T}=T\cup C$ , where $T$ is a plane tree with 3 degree interior vertex and $C$ is a cycle which connects the leaves (vertices of degree 1) of $T$ so that $C$ is the boundary of an infinite face. Except this infinite face, the other faces are called as interior faces. This plane tree $T$ is called the characteristic tree of graph $H_{T}$ . A tree $T$ , whose removal of leaves results in a path $P$ , is called a caterpillar and this path $P$ is called the spine of the caterpillar $T$ . When the characteristic tree of $H_{T}$ is a caterpillar, we call such graph $H_{T}$ cubic caterpillar-Halin graph.

Let $T$ be a caterpillar with spine $P:v_{1},v_{2},\ldots,v_{\ell}$ . We consider the cubic caterpillar-Halin graph $H_{T}.$ In the spine $P,$ each vertex $v_{i}$ is adjacent to a leaf $u_{i}$ for $1\leqslant i\leqslant\ell$ and $v_{1}$ (resp. $v_{\ell}$ ) adjacent to one more leaf $u_{0}=v_{0}$ (resp. $u_{\ell+1}=v_{\ell+1}$ ). We put the path $v_{0}Pv_{\ell+1}$ horizontally in the middle, and then pend these edges $v_{i}u_{i},$ $1\leqslant i\leqslant\ell$ by either up or down edges vertically to $P.$ The graph in Figure 1 is a concrete example for cubic caterpillar-Halin graph.

Counting from the leftmost to the rightmost on $P,$ if the numbers of maximum consecutive up or down edges of the caterpillar-Halin graph $H_{T}$ are $m_{1},m_{2}+1,\cdots,m_{k-1}+1,m_{k}$ , we denote this graph $H_{T}$ by $H_{m_{1},m_{2},\ldots,m_{k}}$ . In order to have a more simple form of the overlap matrix for graph $H_{m_{1},m_{2},\ldots,m_{k}}$ , here we don’t use $m_{1},m_{2},\cdots,m_{k-1},m_{k}$ to denote the numbers of maximum consecutive up or down edges. For example, the graph in Figure 1 is $H_{2,3,3}$ . Figure 2 demonstrates the general case $H_{m_{1},m_{2},\ldots,m_{k}}.$

For convenience, our choice of a spanning tree $T$ for $H_{m_{1},\cdots,m_{k}}$ is indicated by thicker lines. Then, as shown in Figure 2, the co-tree edges are

[TABLE]

If we delete the edges $\{e_{1},\cdots,e_{k}\}$ in graph $H_{m_{1},\cdots,m_{k}}$ , then $m_{1},\cdots,m_{k}$ are the numbers of maximum consecutive co-tree edges actually.

Now, we consider a special case of $H_{m_{1},m_{2},\ldots,m_{k}}.$ When $k=1$ and $m_{1}=m\geqslant 1$ , this graph $H_{m}$ is usually called a Ringel ladder graph denoted by $R_{m}.$ The explicit formula of the embedding distributions for $R_{m}$ was obtained by Chen et al. [3]. Figure 3 demonstrates the graphs $R_{4}$ and $H_{2,2}$ .

In this paper, for any positive integer $m_{1},$ the graph $H_{m_{1},0}$ is defined to be $H_{m_{1}}$ or $R_{m_{1}}$ . For example, the graph $R_{4}$ in Figure 3 is also $H_{4,0}$ .

For any $k\geqslant 2$ and positive integers $m_{1},\cdots,m_{k}$ , the graph $H_{m_{1},\cdots,m_{k},0}$ is defined to be the following graph in Figure 4.

Actually, the graph $H_{m_{1},\cdots,m_{k},0}$ can be interpreted as the graph $H_{m_{1},\cdots,m_{k},m_{k+1}}$ for the special case of $m_{k+1}=0.$

By the definitions above, we observe the following propositions.

Proposition 1.1.

For any positive integers $m_{1},\cdots,m_{k}$ , $H_{m_{1},m_{2},\ldots,m_{k}}\cong H_{m_{k},m_{k-1},\ldots,m_{1}}$

Proposition 1.2.

For any $k\geqslant 2$ and positive integers $m_{1},\cdots,m_{k}$ , we have $H_{m_{1},m_{2},\cdots,m_{k-1},m_{k},0}\cong H_{m_{1},m_{2},\cdots,m_{k-1},m_{k}+1}$ and $H_{m_{1},0}\cong R_{m_{1}}.$

Proposition 1.3.

For any $k\geqslant 2$ and positive integers $m_{1},\cdots,m_{k}$ , we have $H_{m_{1},m_{2},\ldots,m_{k},1}\cong H_{m_{1},m_{2},\ldots,m_{k-1},m_{k}+2}$ and $H_{m_{1},1}\cong R_{m_{1}+1}.$

The graphs $H_{2,2,4}$ and $H_{2,2,3,0}$ in Figure 5 demonstrate Proposition 1.2. The graphes $H_{2,2,2,1}$ and $H_{2,2,4}$ in Figure 5 demonstrate the correction of Proposition 1.3. Actually, all the three graphs in Figure 5 are the same.

1.3 Mohar’s theorem

Let $G$ be a graph with a spanning tree $T$ , $(\rho,\lambda)$ be a $T$ -rotation system and $\beta(G)$ be the Betti number of $G$ . We denote the co-tree edges of $T$ by $e_{1},e_{2},\ldots,e_{\beta(G)}$ . Then, the overlap matrix of $(\rho,\lambda)$ is a matrix $M=[m_{ij}]_{i,j=1}^{\beta(G)}$ over ${\mathbb{Z}}_{2}=\{0,1\}$ with $m_{ij}$ given by

[TABLE]

For $i\neq j,$ when $m_{ij}=1,$ the edges $e_{i}$ and $e_{j}$ are called overlap. Mohar [8] proved that, independent of the choice of spanning tree, the rank of matrix $M$ is the same as the Euler-genus of the corresponding embedding surface.

Let $\varepsilon_{m_{1},\cdots,m_{k}}(j)$ be the number of equivalent embeddings of graph $H_{m_{1},\cdots,m_{k}}$ into surface $S_{j},$ for $j\geqslant 0$ and ${\mathcal{E}}_{m_{1},\cdots,m_{k}}(z)=\displaystyle\sum_{j=0}^{\infty}\varepsilon_{m_{1},\cdots,m_{k}}(j)z^{j}$ be the Euler-genus polynomial of $H_{m_{1},\cdots,m_{k}}.$ We organize this paper as follows: The overlap matrices for graph $H_{m_{1},\cdots,m_{k}}$ are given in Section 2. In Section 3, we obtain a recurrence relation for $\mathcal{E}_{m_{1},\cdots,m_{k}}(z)$ in Theorem 3.4. In Section 4, we derive a recurrence relation for $\mathcal{E}^{1}(t_{1},t_{2},\cdots,t_{k},z),$ where

[TABLE]

In Section 5, some examples are demonstrated to illustrate our results.

2 Overlap matrices for cubic caterpillar-Halin graphs

For a vertex $v$ with degree three in graph $G,$ it has two possibilities. We call them clockwise and counterclockwise. We color the vertex $v$ according to its rotation. If the rotation is clockwise, We color the vertex $v$ black; otherwise, we color it white. Therefore, for any 3-regular graph, we associate their rotation system with a coloring which is called Gustin coloring. In a Gustin coloring, an edge with the same color at both ends is called matched. The other edges are called unmatched.

The following propositions are useful in the calculation of the overlap matrix for $H_{m_{1},\cdots,m_{k}}$ . Applying the Heffter-Edmonds face-tracing algorithm, we can obtain their proof.

Proposition 2.1.

Two co-tree edges overlap if and only if the common edge of the corresponding two interior faces is unmatched.

Proposition 2.2.

The edges $e_{i}$ and $e_{i-1,m_{i-1}}$$(e_{i}$ and $e_{i+1,1})$ overlap, for $i\in\{2,\cdots,k\}$$(i\in\{1,\cdots,k-1\})$ , if and only if the corresponding green edge in Figure 2 is unmatched.

Proposition 2.3.

The edges $e_{i}$ and $e_{i+1}$ overlap, for $i\in\{1,\cdots,k-1\}$ , if and only if the corresponding thicker black edge in Figure 2 is unmatched.

Proposition 2.4.

The edges $e_{i}$ and $e_{i,\ell}$ overlap, for $i\in\{1,\cdots,k\}$ , $1\leqslant\ell\leqslant m_{i}$ if and only if the corresponding red edge in Figure 2 is unmatched.

Proposition 2.5.

The edges $e_{i,\ell}$ and $e_{i,\ell+1}$ overlap, for $i\in\{1,\cdots,k\}$ , $1\leqslant\ell\leqslant m_{i}-1$ if and only if the correspondence blue edge in Figure 2 is unmatched.

For any $n\geqslant m$ and $a=(a_{1},\cdots,a_{n})\in({\mathbb{Z}}_{2})^{n},b=(b_{1},\cdots,b_{n-1})\in({\mathbb{Z}}_{2})^{n-1}$ , we define $\mathcal{L}_{m}^{a,b}$ as a matrix of this form:

[TABLE]

Let ${\mathcal{L}}_{m}=\{\mathcal{L}_{m}^{a,b}:a_{i}\in{\mathbb{Z}}_{2},i=1,\cdots,m;b_{j}\in{\mathbb{Z}}_{2},j=1,\cdots,m-1\}$ and $L_{m}(z)=\sum_{j=0}^{m}L_{m}[j]z^{j},$ where $L_{m}[j]$ is the number of matrices of the form (2.1) for which this matrix has rank $j$ .

Let

[TABLE]

and $X=(\textbf{x}_{0},\cdots,\textbf{x}_{k}),Y=(\textbf{y}_{0},\cdots,\textbf{y}_{k}),Z=(\textbf{z}_{1},\cdots,\textbf{z}_{k})$ . By Propositions 2.1-2.5, for any $m_{1}\geqslant 1,\cdots,m_{k}\geqslant 1$ , the overlap matrix of $H_{m_{1},\cdots,m_{k}}$ has this form:

[TABLE]

where $M_{1}$ is a $m_{1}\times k$ matrix given by

[TABLE]

For $2\leqslant i\leqslant k-1$ , $M_{i}$ is a $m_{i}\times k$ matrix with the $i-1,i,i+1$ columns given by the following three columns, respectively,

[TABLE]

the other columns of $M_{i}$ are $\textbf{0};$ $M_{k}$ is a $m_{k}\times k$ matrix given by

[TABLE]

In this paper, the transpose of $M_{i}$ is denoted by $M_{i}^{T}$ .

We can also write $\Delta_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ in the this form:

[TABLE]

Actually, for $i\in\{1,\cdots,k\}$ ,

•

$x_{i}=1$ if and only if the edge $e_{i}$ is twisted.

•

$1\leqslant\ell\leqslant m_{i}$ , $x_{i,\ell}=1$ if and only if the edge $e_{i,\ell}$ is twisted.

For $i\in\{2,\cdots,k\}$ ,

•

$z_{i,0}=1$ if and only if the edges $e_{i}$ and $e_{i-1,m_{i-1}}$ overlap.

For $i\in\{1,\cdots,k-1\}$ ,

•

$z_{i,m_{i}+1}=1$ if and only if the edges $e_{i}$ and $e_{i+1,1}$ overlap,

•

$y_{i}=1$ if and only if the edges $e_{i}$ and $e_{i+1}$ overlap.

For $i\in\{1,\cdots,k\}$ and

•

$1\leqslant\ell\leqslant m_{i}$ , $z_{i,\ell}=1$ if and only if the edges $e_{i}$ and $e_{i,\ell}$ overlap,

•

$1\leqslant\ell\leqslant m_{i}-1$ , $y_{i,\ell}=1$ if and only if the edges $e_{i,\ell}$ and $e_{i,\ell+1}$ overlap.

Theorem 2.6.

Consider the cubic caterpillar-Halin graph $H_{m_{1},\cdots,m_{k}}$ . For any fixed overlap matrix $\Delta_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ of graph $H_{m_{1},\cdots,m_{k}}$ , there are exactly 2 different $T$ -rotation systems of $H_{m_{1},\cdots,m_{k}}$ corresponding to that matrix.

Proof.

The values of $\textbf{y}_{0},\cdots,\textbf{y}_{k},\textbf{z}_{1},\cdots,\textbf{z}_{k}$ in (2.2) can be determined according to the matrix $\Delta_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ . We make the following discussions.

•

Case 1: $z_{1,1}=1$ . If we color the vertex $v_{0,0}$ in Figure 2 black, by Propositions 2.1,2.4, we need to color the vertex $v_{1,1}$ white. Since the values of $\textbf{y}_{0},\cdots,\textbf{y}_{k},\textbf{z}_{1},\cdots,\textbf{z}_{k}$ are given, by recursions and Propositions 2.1-2.5, we color all the rest of vertices. Thus, all the rotations of vertices in $H_{m_{1},\cdots,m_{k}}$ are determined. For the case we color the vertex $v_{0,0}$ white, all the rotations of vertices in $H_{m_{1},\cdots,m_{k}}$ are determined by a similar way.

•

Case 2: $z_{1,1}=0$ . The discussions are similar and we omit the details.

Since we can color the rooted vertex black or white, we finish our proof. ∎

Let $\delta_{m_{1},m_{2},\cdots,m_{k}}(z)=\sum_{j=0}^{k+m_{1}+\cdots+m_{k}}\delta_{m_{1},m_{2},\cdots,m_{k}}[j]z^{j}$ , where $\delta_{m_{1},m_{2},\cdots,m_{k}}[j]$ is the number of matrices of the form $\Delta_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ for which this matrix has rank $j$ . By Theorem 2.6, we have the following relation between $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ and $\delta_{m_{1},\cdots,m_{k}}(z).$

Theorem 2.7.

We have

[TABLE]

where $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ is the Euler-genus polynomial of $H_{m_{1},\cdots,m_{k}}$ .

For $X=(\textbf{x}_{0},\cdots,\textbf{x}_{k}),Y=(\textbf{y}_{0},\cdots,\textbf{y}_{k}),Z=(\textbf{z}_{1},\cdots,\textbf{z}_{k-1})$ , when the $k$ -th row and $k$ -th column of matrix $\Delta_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ are $\textbf{0},$ we denote this matrix by $\Lambda_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ . We can also write the matrix $\Lambda_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ in the following form.

[TABLE]

Actually, $\Lambda_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ is the overlap matrix for graph $H_{n_{1},\cdots,n_{k}}-e_{k}$ which is obtained by deleting the edge $e_{k}$ in the graph $H_{n_{1},\cdots,n_{k}}$ .

For any $m_{1}\geqslant 1,\cdots,m_{k}\geqslant 1$ , we define the following polynomial related to $\Lambda_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ ,

[TABLE]

where $\lambda_{m_{1},m_{2},\cdots,m_{k}}[j]$ is the number of matrices of the form $\Lambda_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ for which this matrix has rank $j$ .

In [3], Chen et al. obtained the following overlap matrix for Ringel ladder $R_{m-1}$ :

[TABLE]

where $\textbf{x}=(x_{0},x_{1},\cdots,x_{m})\in{\mathbb{Z}}_{2}^{m+1},\textbf{y}=(y_{1},\cdots,y_{m-1})\in{\mathbb{Z}}_{2}^{m-1}$ , $\textbf{z}=(z_{1},\cdots,z_{m-1},z_{m})\in{\mathbb{Z}}_{2}^{m}$ . Let

[TABLE]

and $\phi_{m+1}(z)=\sum_{j=0}^{m+1}\phi_{m+1}[j]z^{j}$ , where $\phi_{m+1}[j]$ is the number of matrices in $\Phi_{m+1}$ with rank $j$ .

By the fact $H_{m_{1},0}\cong R_{m_{1}}$ and Proposition 1.1-Proposition 1.3, one sees that

[TABLE]

In this article, the following previously derived results are useful.

Proposition 2.8.

*([2])

The following recurrence relation holds for $L_{n}(z),n\geqslant 3$ ,*

[TABLE]

and its initial conditions are given by $L_{1}(z)=z+1,L_{2}(z)=4z^{2}+3z+1.$

Proposition 2.9.

*([3])

The following recurrence relation holds for $\phi_{m}(z),m\geqslant 3$ *

[TABLE]

with initial conditions $\phi_{2}(z)=4z^{2}+3z+1,\phi_{3}(z)=28z^{3}+28z^{2}+7z+1.$ Furthermore, we have

[TABLE]

3 Euler-genus polynomials $\mathcal{E}_{m_{1},\cdots,m_{k}}(z)$ for graph $H_{m_{1},\cdots,m_{k}}$

In subsections 3.1 and 3.2, we give the computations of $\mathcal{E}_{{m_{1},m_{2}}}(z)$ and $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ respectively.

3.1 The computation of ${\mathcal{E}}_{{m_{1},m_{2}}}(z)$ .

By definition, $\Delta_{m_{1},m_{2}}^{X,Y,Z}$ is a matrix given by

[TABLE]

By definition, the matrix $\Lambda_{m_{1},m_{2}}^{X,Y,Z}$ is given by

[TABLE]

Now, we give a lemma to demonstrate the recurrence relation for polynomial $\lambda_{m_{1},m_{2}}(z)$ .

Lemma 3.1.

For any $m_{1}\geqslant 1,m_{2}\geqslant 2$ , the following recurrence relation holds for $\lambda_{m_{1},m_{2}}(z)$

[TABLE]

with initial conditions $\lambda_{m_{1},0}(z)=\phi_{m_{1}+1}(z),\lambda_{m_{1},1}(z)=(1+2z)\phi_{m_{1}+1}(z)+2^{m_{1}+1}z^{2}L_{m_{1}}(z)$

Proof.

There are $4$ different choices of values for $(x_{2,m_{2}},y_{2,m_{2}-1}),m_{2}\geqslant 3.$

[TABLE]

Combining cases 1-4, one arrives at (3.1).

By direct computation, one sees that

[TABLE]

So, this theorem holds with $m_{2}=2.$

∎

We have the recurrence relation for ${\mathcal{E}}_{{m_{1},m_{2}}}(z)$ .

Theorem 3.2.

For any $m_{1}\geqslant 1,m_{2}\geqslant 2$ , the following recurrence relation holds for ${\mathcal{E}}_{{m_{1},m_{2}}}(z)$ ,

[TABLE]

with initial conditions ${\mathcal{E}}_{{m_{1},0}}(z)=2\phi_{m_{1}+2}(z),{\mathcal{E}}_{{m_{1},1}}(z)=2\phi_{m_{1}+3}(z).$

Proof.

The initial conditions come from (2.4) and Theorem 2.7.

Consider the overlap matrix $\Delta_{m_{1},m_{2}}^{X,Y,Z}$ of graph $H_{m_{1},m_{2}}$ . Since there are eight different choices of values for $(x_{2,m_{2}},y_{2,m_{2}-1},z_{2,m_{2}})$ ,

[TABLE]

one arrives at

[TABLE]

Combing the above equality with Theorem 2.7, the proof is completed. ∎

3.2 The computation of $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ .

Before we give a recurrence relation for $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ , we introduce a lemma.

Lemma 3.3.

The following recurrence relation holds for the polynomial $\lambda_{m_{1},m_{2},\cdots,m_{k}}(z)$ $(k\geqslant 3,m_{1},\cdots,m_{k-1}\geqslant 1,m_{k}\geqslant 2)$ ,

[TABLE]

with the initial conditions

[TABLE]

Proof.

There are $4$ different combinations of values for the variables $(x_{k,m_{k}},y_{k,m_{k}-1}):$

[TABLE]

Combining cases 1-4, (3.3) holds with $m_{k}\geqslant 3$ .

By direct calculating, one sees that

[TABLE]

which implies (3.3) holds with $m_{k}=2$ , that is the initial conditions hold. ∎

We have the following recurrence relation for $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ .

Theorem 3.4.

The polynomial $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)(m_{1},\cdots,m_{k-1}\geqslant 1,m_{k}\geqslant 2)$ satisfies the recurrence relation

[TABLE]

with the initial conditions $\mathcal{E}_{{m_{1},\cdots,m_{k-1},0}}(z)=\mathcal{E}_{{m_{1},\cdots,m_{k-1}+1}}(z),\mathcal{E}_{{m_{1},\cdots,m_{k-1},1}}(z)=\mathcal{E}_{{m_{1},\cdots,m_{k-1}+2}}(z).$

Before the proof of Theorem 3.4, we explain here how to calculate $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ .

Remark 3.5.

Fix $m_{1},\cdots,m_{k}.$ We will use iteration to calculate

[TABLE]

First, in Lemma 3.1 and Theorem 3.2, we get the values of

[TABLE]

Assume that we have already get the values of (3.5) for $k\leqslant\ell-1.$ Using Lemma 3.3, the values of $\lambda_{m_{1},\cdots,m_{\ell-1},i}(z),0\leqslant i\leqslant m_{\ell}$ are obtained. Then, by Theorem 3.4, one arrives at the values of

[TABLE]

By the above arguments, we can get the explicit expression for ${\mathcal{E}}_{m_{1},\cdots,m_{k-1},m_{k}}(z)$ .

3.3 Proof of Theorem 3.4

Proof.

Noting $H_{m_{1},\cdots,m_{k-1},0}=H_{m_{1},\cdots,m_{k-1}+1}$ and Proposition1.1-Proposition 1.3, one sees that the initial conditions hold. So, we only need to prove (3.4).

Consider the overlap matrix $\Delta_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ of graph $H_{m_{1},\cdots,m_{k}}$ . There are the following eight different combinations of values for the variables $x_{k,m_{k}},y_{k,m_{k}-1}$ and $z_{k,m_{k}}$ .

Case 1: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(0,0,0)$ . One easily sees that this case contributes to $\delta_{m_{1},m_{2},\cdots,m_{k}}(z)$ by a term $\delta_{m_{1},m_{2},\cdots,m_{k}-1}(z).$

Case 2: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(0,0,1)$ . If $z_{k,0}=1,$ we add row $-1$ to row $-(m_{k}+1)$ and then add column $-1$ to column $-(m_{k}+1)$ . Here row(column) $-\ell$ denotes the row $\ell$ counting from the last row(column). Making similar operations for $z_{k,1},\cdots,z_{k,m_{k}-1}$ and $x_{k},y_{k-1},$ the resulting matrix is given by

[TABLE]

Since there are $2^{m_{k}+2}$ choices of values for $z_{k,0},\cdots,z_{k,m_{k}-1}$ and $x_{k},y_{k-1}$ , this case contributes a term $2^{m_{k}+2}z^{2}\lambda_{m_{1},m_{2},\cdots,m_{k-1},m_{k}-1}(z).$

Case 3: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(1,0,0)$ . This case contributes a term $z\delta_{m_{1},m_{2},\cdots,m_{k}-1}(z).$

Case 4: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(0,1,0)$ . If $z_{k,m_{k}-1}=1,$ we add row $-1$ to row $k$ , then we add the column $-1$ to the column $k$ . We make similar operations for $x_{k,m_{k}-1},y_{k,m_{k}-2}$ and get the following matrix.

[TABLE]

Since there are $2^{3}$ different choices of values for the variables $x_{k,m_{k}-1},y_{k,m_{k}-2},z_{k,m_{k}-1},$ it contributes a term $8z^{2}\delta_{m_{1},m_{2},\cdots,m_{k-1},m_{k}-2}(z).$

Case 5: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(0,1,1)$ . We add row $-2$ to row $k$ and add column $-2$ to column $k$ . We get the following matrix.

[TABLE]

where $z_{k,m_{k}-2}^{*}=z_{k,m_{k}-2}+y_{k,m_{k}-2}$ and $z_{k,m_{k}-1}^{*}=z_{k,m_{k}-1}+x_{k,m_{k}-1}.$ If $y_{k,m_{k}-2}=1$ , we add row $-1$ to row $-3$ and then add column $-1$ to column $-3$ . We make similar discussions for $x_{k,m_{k}-1},z_{k,m_{k}-1}^{*}$ and transform the matrix $\Delta_{m_{1},\cdots,m_{k}}^{X,Y,Z}$ into the following form

[TABLE]

Since there are $2^{3}$ possible choices of values for $x_{k,m_{k}-1},y_{k,m_{k}-2},z_{k,m_{k}-1}$ , this case contributes a term $8z^{2}\delta_{m_{1},m_{2},\cdots,m_{k-1},m_{k}-2}(z).$

Case 6: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(1,1,0)$ . We add row $-1$ to row $-2$ and add column $-1$ to column $-2$ . Then, one sees that it contributes a term $z\delta_{m_{1},m_{2},\cdots,m_{k-1},m_{k}-1}(z)$ .

Case 7: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(1,0,1)$ . We add row $-1$ to row $k$ and then add column $-1$ to column $k$ . it contributes a term $z\delta_{m_{1},m_{2},\cdots,m_{k-1},m_{k}-1}(z)$ .

Case 8: $(x_{k,m_{k}},y_{k,m_{k}-1},z_{k,m_{k}})=(1,1,1)$ . We add row $-1$ to the rows $-2,k$ and then add column $-1$ to the columns $-2,k$ respectively. Then, it contributes a term $z\delta_{m_{1},m_{2},\cdots,m_{k-1},m_{k}-1}(z)$ .

Combining cases 1-8, we obtain

[TABLE]

Combing the above equality with Theorem 2.7, one gets (3.4). ∎

Recall that the definition of $\varepsilon_{m_{1},\cdots,m_{k}}(j)$ is given in subsection 1.3.

Theorem 3.6.

For any $k\geqslant 2$ and positive integers $m_{1},\cdots,m_{k},$ we have $\varepsilon_{m_{1},\cdots,m_{k}}(0)=2$ and $\varepsilon_{m_{1},\cdots,m_{k}}(1)=8(m_{1}+m_{2}+\cdots+m_{k}+k)-10.$

Proof.

For any $m_{1},\cdots,m_{k}\geqslant 1,$ obviously, we have $\varepsilon_{m_{1},\cdots,m_{k}}(0)=2.$ We use induction on $k$ to prove

[TABLE]

First, we claim that, for $k=2,$ (3.6) holds. By Proposition 2.9, one sees that

[TABLE]

which yields

[TABLE]

Furthermore, using Theorem 3.2, we have

[TABLE]

which implies that $\varepsilon_{m_{1},m_{2}}(1)=\varepsilon_{m_{1},1}(1)+8(m_{2}-1)=8m_{1}+14+8(m_{2}-1)=8(m_{1}+m_{2}+2)-10.$ Therefore, (3.6) holds with $k=2.$

Assume (3.6) holds for any $m_{1},\cdots,m_{k-1}\geqslant 1$ . By Theorem 3.4, one easily sees that

[TABLE]

Furthermore, by induction, we have

[TABLE]

which completes the proof of (3.6). ∎

With a similar method, one has the following results.

Theorem 3.7.

For any $k\geqslant 2$ and $m_{1},\cdots,m_{k}\geqslant 1,$ $\varepsilon_{m_{1},\cdots,m_{k}}(2)=8\sum_{i=1}^{k}\Big{(}-9+4i+(-3+4i)m_{i}+2m_{i}^{2}\Big{)}+2\sum_{i=2}^{k}(2^{m_{i}+3}+3\cdot 2^{m_{i-1}+3})-2^{m_{1}+5}+32\sum_{i=2}^{k}\sum_{j=1}^{i-1}(m_{j}m_{i}+m_{j}).$

4 The generating functions for graphs $H_{m_{1},\cdots,m_{k}}.$

For $i=0,1,2,3$ , we define

[TABLE]

and $L^{*}(t,z)=\sum_{m\geqslant 1}2^{m}L_{m}(z)t^{m}.$

In subsection 4.1, we give the computations of $\lambda^{1}(t_{1},t_{2},z),\lambda^{2}(t_{1},t_{2},z),\mathcal{E}^{1}(t_{1},t_{2},z),$ and $\mathcal{E}^{2}(t_{1},t_{2},z)$ . In subsection 4.2, we list some lemmas. These lemmas demonstrate the relations between

[TABLE]

Using these lemmas in subsections 4.1,4.2, we compute the following functions in subsection 4.3,

[TABLE]

4.1 The computations of $\lambda^{1}(t_{1},t_{2},z),\lambda^{2}(t_{1},t_{2},z),\mathcal{E}^{1}(t_{1},t_{2},z),\mathcal{E}^{2}(t_{1},t_{2},z)$

First, one easily sees the following lemma hold.

Lemma 4.1.

one has $L^{*}(t,z)=\frac{8z^{2}t^{2}+2t(1+z)}{-16z^{2}t^{2}-(4z+2)t+1}.$

The expressions of $(\lambda^{1}(t_{1},t_{2},z),\lambda^{2}(t_{1},t_{2},z))$ and $(\mathcal{E}^{1}(t_{1},t_{2},z),\mathcal{E}^{2}(t_{1},t_{2},z))$ are given by the following two lemmas .

Lemma 4.2.

We have

[TABLE]

where the function $\phi$ is given by Proposition 2.9.

Proof.

By Lemma 3.1, we have

[TABLE]

That is

[TABLE]

Therefore, by Lemma 3.1, one sees that

[TABLE]

Now, we consider the computation of $\lambda^{2}(t_{1},t_{2},z)$ . By Lemma 3.1, it holds that

[TABLE]

∎

Lemma 4.3.

We have

[TABLE]

Proof.

By Theorem 3.2, one sees that

[TABLE]

On the other hand, it also holds that

[TABLE]

Combining (4.2) with (4.1), we finish the proof. ∎

4.2 Some lemmas

In this subsection, we list some lemmas used in subsection 4.3.

Lemma 4.4.

For $k\geqslant 3,$ we have $\lambda^{1}(t_{1},t_{2},\cdots,t_{k},z)=\lambda^{0}(t_{1},t_{2},\cdots,t_{k},z)-\frac{\mathcal{E}^{1}(t_{1},t_{2},\cdots,t_{k-1},z)}{2}.$

Proof.

By Lemma 3.3, we obtain

[TABLE]

∎

Lemma 4.5.

For any $k\geqslant 3,$ we have $\lambda^{2}(t_{1},t_{2},\cdots,t_{k},z)=\lambda^{1}(t_{1},t_{2},\cdots,t_{k},z)-t_{k}(1+2z)\mathcal{E}^{1}(t_{1},\cdots$ $,t_{k-1},z)-16z^{2}t_{k}\lambda^{1}(t_{1},\cdots,t_{k-1,z}).$

Proof.

With the help of Lemma 3.3, we obtain

[TABLE]

∎

Lemma 4.6.

For any $k\geqslant 3,$ we have

[TABLE]

Proof.

Using (3.3), we have

[TABLE]

That is $\lambda^{2}(t_{1},t_{2},\cdots,t_{k},z)=(2+4z)t_{k}\lambda^{1}(t_{1},t_{2},\cdots,t_{k},z)+16z^{2}t_{k}^{2}\lambda^{0}(t_{1},t_{2},\cdots,t_{k},z).$

∎

Lemma 4.7.

For any $k\geqslant 3,$ we have

[TABLE]

Proof.

By Theorem 3.4, we have

[TABLE]

which completes the proof. ∎

Lemma 4.8.

For $k\geqslant 3,$ one has

[TABLE]

Proof.

By Theorem 3.4, one arrives at that

[TABLE]

∎

Lemma 4.9.

We have $\mathcal{E}^{1}(t_{1},t_{2},\cdots,t_{k},z)=\mathcal{E}^{0}(t_{1},t_{2},\cdots,t_{k},z)-t_{k-1}^{-1}\mathcal{E}^{2}(t_{1},t_{2},\cdots,t_{k-1},z).$

Proof.

Noting these initial conditions in Theorem 3.4, one easily sees that

[TABLE]

∎

Lemma 4.10.

We have $\mathcal{E}^{2}(t_{1},\cdots,t_{k},z)=\mathcal{E}^{1}(t_{1},\cdots,t_{k},z)-t_{k}t_{k-1}^{-2}\mathcal{E}^{3}(t_{1},\cdots,t_{k-1},z).$

Proof.

By these initial conditions in Theorem 3.4, we obtain

[TABLE]

∎

4.3 The calculation of generating functions

In this subsection, we will demonstrate a recurrence relation between

[TABLE]

and

[TABLE]

In achieve them, we demonstrate two lemmas first.

Lemma 4.11.

For $k\geqslant 3,$ we have

[TABLE]

where

[TABLE]

Proof.

By Lemmas 4.4, 4.6, one sees that

[TABLE]

Combining the above equality with Lemma 4.5, we get

[TABLE]

which completes the proof of (4.3). Using Lemma 4.5 again, we get (4.4). ∎

Lemma 4.12.

For $k\geqslant 3,$ we have

[TABLE]

and

[TABLE]

*where *

[TABLE]

Proof.

With a help of Lemma 4.7 and Lemma 4.9, one sees

[TABLE]

Using Lemmas 4.8, 4.10, we obtain

[TABLE]

Combining (4.7) with (4.3) (4.8), one arrives at that

[TABLE]

which finishes the proof of (4.5). Combining (4.5) with (4.8), one arrives at (4.6). ∎

Now, we state our main results in this section

Theorem 4.13.

One has

[TABLE]

where $A(t_{k-1},t_{k},z)$ is a matrix given by

[TABLE]

And furthermore, we have

[TABLE]

where the values of $\lambda^{i}(t_{1},t_{2},z),\mathcal{E}^{i}(t_{1},t_{2},z)$ are given in subsection 4.2.

For any $k$ , by Theorem 4.13, one can derive the expression of $\mathcal{E}^{1}(t_{1},t_{2},\cdots,t_{k},z).$ Since

[TABLE]

we can obtain the expression of $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ for any fixed $m_{1}\geqslant 1,\cdots,m_{k}\geqslant 1$ .

5 Examples

For any $m_{1}\geqslant 1,\cdots,m_{k}\geqslant 1,$ by the results in Section 4, one gets the expression of $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ . We give two examples to demonstrate this.

Example 5.1.

Let $k=2$ . We have

[TABLE]

Example 5.2.

Let $k=3$ . We have

[TABLE]

$\bf{Acknowledgement}$ We greatly appreciate professor Yichao Chen’s useful discussion.

Bibliography11

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] J. Chen, J. L. Gross, and R. G. Rieper, Overlap matrices and total embeddings, Discrete Mathematics 128 (1994), 73-94.
2[2] Y. Chen, T.Mansour, and Q. Zou, Embedding distributions and chebyshev polynomials, Graphs and Combinatorics 28 (2012) 597-614.
3[3] Y. Chen, L. Ou, and Q. Zou, Total embedding distributions of Ringel ladders, Discrete Mathematics . 311 (2011) 2463-2474.
4[4] Y. Chen and J.L. Gross, An Euler-genus approach to the calculation of crosscap-number polynomial, Journal of Graph Theory 88 (2018) 80–100.
5[5] Y. Chen, J.L. Gross, and T. Mansour, On the genus distributions of wheels and of related graphs, Discrete Mathematics 341 (2018) 934–945.
6[6] J. L. Gross, Embeddings of cubic Halin graphs: Genus distributions, Ars Mathematica Contemporanea . 6 (2011),37-56.
7[7] E. Enami, Inequivalent embeddings of 3-connected 3-regular planar graphs on the torus, ar Xiv:1806.11333, 2018.
8[8] B. Mohar, An obstruction to embedding graphs in surfaces, Discrete Mathematics . 78 (1989), 135-142.

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Euler-genus distributions of cubic caterpillar-Halin graphs

Abstract

1 Introduction

1.1 Euler-genus polynomial

1.2 Cubic caterpillar-Halin graphs

Proposition 1.1**.**

Proposition 1.2**.**

Proposition 1.3**.**

1.3 Mohar’s theorem

2 Overlap matrices for cubic caterpillar-Halin graphs

Proposition 2.1**.**

Proposition 2.2**.**

Proposition 2.3**.**

Proposition 2.4**.**

Proposition 2.5**.**

Theorem 2.6**.**

Proof.

Theorem 2.7**.**

Proposition 2.8**.**

Proposition 2.9**.**

3 Euler-genus polynomials Em1,⋯ ,mk(z)\mathcal{E}_{m_{1},\cdots,m_{k}}(z)Em1​,⋯,mk​​(z) for graph Hm1,⋯ ,mkH_{m_{1},\cdots,m_{k}}Hm1​,⋯,mk​​

3.1 The computation of Em1,m2(z){\mathcal{E}}_{{m_{1},m_{2}}}(z)Em1​,m2​​(z).

Lemma 3.1**.**

Proof.

Theorem 3.2**.**

Proof.

3.2 The computation of Em1,⋯ ,mk(z)\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)Em1​,⋯,mk​​(z).

Lemma 3.3**.**

Proof.

Theorem 3.4**.**

Remark 3.5**.**

3.3 Proof of Theorem 3.4

Proof.

Theorem 3.6**.**

Proof.

Theorem 3.7**.**

4 The generating functions for graphs Hm1,⋯ ,mk.H_{m_{1},\cdots,m_{k}}.Hm1​,⋯,mk​​.

4.1 The computations of λ1(t1,t2,z),λ2(t1,t2,z),E1(t1,t2,z),E2(t1,t2,z)\lambda^{1}(t_{1},t_{2},z),\lambda^{2}(t_{1},t_{2},z),\mathcal{E}^{1}(t_{1},t_{2},z),\mathcal{E}^{2}(t_{1},t_{2},z)λ1(t1​,t2​,z),λ2(t1​,t2​,z),E1(t1​,t2​,z),E2(t1​,t2​,z)

Lemma 4.1**.**

Lemma 4.2**.**

Proof.

Lemma 4.3**.**

Proof.

4.2 Some lemmas

Lemma 4.4**.**

Proof.

Lemma 4.5**.**

Proof.

Lemma 4.6**.**

Proof.

Lemma 4.7**.**

Proof.

Lemma 4.8**.**

Proof.

Lemma 4.9**.**

Proof.

Lemma 4.10**.**

Proof.

4.3 The calculation of generating functions

Lemma 4.11**.**

Proof.

Lemma 4.12**.**

Proof.

Theorem 4.13**.**

5 Examples

Example 5.1**.**

Example 5.2**.**

Proposition 1.1.

Proposition 1.2.

Proposition 1.3.

Proposition 2.1.

Proposition 2.2.

Proposition 2.3.

Proposition 2.4.

Proposition 2.5.

Theorem 2.6.

Theorem 2.7.

Proposition 2.8.

Proposition 2.9.

3 Euler-genus polynomials $\mathcal{E}_{m_{1},\cdots,m_{k}}(z)$ for graph $H_{m_{1},\cdots,m_{k}}$

3.1 The computation of ${\mathcal{E}}_{{m_{1},m_{2}}}(z)$ .

Lemma 3.1.

Theorem 3.2.

3.2 The computation of $\mathcal{E}_{{m_{1},\cdots,m_{k}}}(z)$ .

Lemma 3.3.

Theorem 3.4.

Remark 3.5.

Theorem 3.6.

Theorem 3.7.

4 The generating functions for graphs $H_{m_{1},\cdots,m_{k}}.$

4.1 The computations of $\lambda^{1}(t_{1},t_{2},z),\lambda^{2}(t_{1},t_{2},z),\mathcal{E}^{1}(t_{1},t_{2},z),\mathcal{E}^{2}(t_{1},t_{2},z)$

Lemma 4.1.

Lemma 4.2.

Lemma 4.3.

Lemma 4.4.

Lemma 4.5.

Lemma 4.6.

Lemma 4.7.

Lemma 4.8.

Lemma 4.9.

Lemma 4.10.

Lemma 4.11.

Lemma 4.12.

Theorem 4.13.

Example 5.1.

Example 5.2.