Chapter Questions
Find a Hamiltonian circuit in the digraph given in Example 7 different from the one in Figure $30.3$.
Find a Hamiltonian circuit in$$\operatorname{Cay}\left(\{(a, 0),(b, 0),(e, 1)\}: Q_{4} \oplus Z_{2}\right)$$
Find a Hamiltonian circuit in$$\operatorname{Cay}\left(\{(a, 0),(b, 0),(e, 1)\}: Q_{4} \oplus Z_{m}\right)$$where $m$ is even.
Write the sequence of generators for each of the circuits found in Exercises 1,2, and $3 .$
Use the Cayley digraph in Example 7 to evaluate the product $a^{3} b a^{-1} b a^{3} b^{-1}$
Let $x$ and $y$ be two vertices of a Cayley digraph. Explain why two paths from $x$ to $y$ in the digraph yield a group relation $-$ that is, an equation of the form $a_{1} a_{2} \cdots a_{m}=b_{1} b_{2} \cdots b_{n}$, where the $a_{i}$ 's and $b_{j}$ 's are generators of the Cayley digraph.
Use the Cayley digraph in Example 7 to verify the relation $a b a^{-1} b^{-1} a^{-1} b^{-1}=a^{2} b a^{3}$.
Identify the following Cayley digraph of a familiar group.
Let $D_{4}=\left\langle r, f \mid r^{4}=e=f^{2}, r f=f r^{-1}\right\rangle$. Verify that$$6 *[3 *(r, 0),(f, 0), 3 *(r, 0),(e, 1)]$$is a Hamiltonian circuit in$$\operatorname{Cay}\left(\{(r, 0),(f, 0),(e, 1)\}: D_{4} \oplus Z_{6}\right)$$
Draw a picture of $\operatorname{Cay}\left(\{2,5\}: Z_{8}\right)$.
If $s_{1}, s_{2}, \ldots, s_{n}$ is a sequence of generators that determines a Hamiltonian circuit beginning at some vertex, explain why the same sequence determines a Hamiltonian circuit beginning at any point. (This exercise is referred to in this chapter.)
Show that the Cayley digraph given in Example 7 has a Hamiltonian path from $e$ to $a$.
Show that there is no Hamiltonian path in$$\operatorname{Cay}\left(\{(1,0),(0,1)\}: Z_{3} \oplus Z_{2}\right)$$from $(0,0)$ to $(2,0)$
Draw $\operatorname{Cay}\left(\{2,3\}: Z_{6}\right)$. Is there a Hamiltonian circuit in this digraph?
a. Let $G$ be a group of order $n$ generated by a set $S$. Show that a sequence $s_{1}, s_{2}, \ldots, s_{n-1}$ of elements of $S$ is a Hamiltonian path in $\operatorname{Cay}(S: G)$ if and only if, for all $i$ and $j$ with $1 \leq i \leq j<n$, we have $s_{i} s_{i+1} \cdots s_{j} \neq e$b. Show that the sequence $s_{1} s_{2} \cdots s_{n}$ is a Hamiltonian circuit if and only if $s_{1} s_{2} \cdots s_{n}=e$, and that whenever $1 \leq i \leq j<n$, we have $s_{i} s_{i+1} \cdots s_{j} \neq e$
Let $D_{4}=\left\langle a, b \mid a^{2}=b^{2}=(a b)^{4}=e\right\rangle .$ Draw $\operatorname{Cay}\left(\{a, b\}: D_{4}\right)$. Why is it reasonable to say that this digraph is undirected?
Let $D_{n}$ be as in Example $10 .$ Show that $2 *[(n-1) * r, f]$ is a Hamiltonian circuit in $\operatorname{Cay}\left(\{r, f\}: D_{n}\right)$.
Let $Q_{8}=\left\langle a, b \mid a^{8}=e, a^{4}=b^{2}, b^{-1} a b=a^{-1}\right\rangle .$ Find a Hamiltonian circuit in $\operatorname{Cay}\left(\{a, b\}: Q_{8}\right)$.
Let $Q_{8}$ be as in Exercise 18 . Find a Hamiltonian circuit in$$\operatorname{Cay}\left(\{(a, 0),(b, 0),(e, 1)\}: Q_{8} \oplus Z_{5}\right)$$
Prove that the Cayley digraph given in Example 6 does not have a Hamiltonian circuit. Does it have a Hamiltonian path?
Find a Hamiltonian circuit in$$\operatorname{Cay}\left(\left\{\left(R_{90}, 0\right),(H, 0),\left(R_{0}, 1\right)\right\}: D_{4} \oplus Z_{3}\right)$$Does this circuit generalize to the case $D_{n+1} \oplus Z_{n}$ for all $n \geq 3$ ?
Let $Q_{8}$ be as in Exercise $18 .$ Find a Hamiltonian circuit in $\operatorname{Cay}\left(\{(a, 0),(b, 0),(e, 1)\}: Q_{8} \oplus Z_{m}\right)$ for all even $m$
Find a Hamiltonian circuit in$$\operatorname{Cay}\left(\{(a, 0),(b, 0),(e, 1)\}: Q_{4} \oplus Z_{3}\right)$$
Find a Hamiltonian circuit in $\operatorname{Cay}\left(\{(a, 0),(b, 0),(e, 1)\}: Q_{4} \oplus Z_{m}\right)$ for all odd $m \geq 3$
Write the sequence of generators that describes the Hamiltonian circuit in Example $9 .$
Let $D_{n}$ be as in Example 10 . Find a Hamiltonian circuit in$$\operatorname{Cay}\left(\{(r, 0),(f, 0),(e, 1)\}: D_{4} \oplus Z_{5}\right)$$Does your circuit generalize to the case $D_{n} \oplus Z_{n+1}$ for all $n \geq 4 ?$
Prove that $\operatorname{Cay}\left(\{(0,1),(1,1)\}: Z_{m} \oplus Z_{n}\right)$ has a Hamiltonian circuit for all $m$ and $n$ greater than 1 .
Suppose that a Hamiltonian circuit exists for $\operatorname{Cay}(\{(1,0),(0,1)\}$ : $\left.Z_{m} \oplus Z_{n}\right)$ and that this circuit exits from vertex $(a, b)$ vertically. Show that the circuit exits from every member of the coset $(a, b)+\langle(1,-1)\rangle$ vertically.
Let $D_{2}=\left\langle r, f \mid r^{2}=f^{2}=e, r f=f r^{-1}\right\rangle .$ Find a Hamiltonian circuit in $\operatorname{Cay}\left(\{(r, 0),(f, 0),(e, 1)\}: D_{2} \oplus Z_{3}\right)$.
Let $Q_{8}$ be as in Exercise 18 . Find a Hamiltonian circuit in $\operatorname{Cay}(\{(a, 0)$, $\left.(b, 0),(e, 1)\}: Q_{8} \oplus Z_{3}\right)$
In $\operatorname{Cay}\left(\{(1,0),(0,1)\}: Z_{4} \oplus Z_{5}\right)$, find a sequence of generators that visits exactly one vertex twice and all others exactly once and returns to the starting vertex.
In $\operatorname{Cay}\left(\{(1,0),(0,1)\}: Z_{4} \oplus Z_{5}\right)$, find a sequence of generators that visits exactly two vertices twice and all others exactly once and returns to the starting vertex.
Find a Hamiltonian circuit in $\operatorname{Cay}\left(\{(1,0),(0,1)\}: Z_{4} \oplus Z_{6}\right)$.
Let $G$ be the digraph obtained from $\operatorname{Cay}\left(\{(1,0),(0,1)\}: Z_{3} \oplus Z_{5}\right)$ by deleting the vertex $(0,0) .$ [Also, delete each arc to or from $(0,0) .]$ Prove that $G$ has a Hamiltonian circuit.
Prove that the digraph obtained from $\operatorname{Cay}\left(\{(1,0),(0,1)\}: Z_{4} \oplus Z_{7}\right)$ by deleting the vertex $(0,0)$ has a Hamiltonian circuit.
Let $G$ be a finite group generated by $a$ and $b$. Let $s_{1}, s_{2}, \ldots, s_{n}$ be the arcs of a Hamiltonian circuit in the digraph $\operatorname{Cay}(\{a, b\}: G)$. We say that the vertex $s_{1} s_{2} \cdots s_{i}$ travels by $a$ if $s_{i+1}=a .$ Show that if a vertex $x$ travels by $a$, then every vertex in the coset $x\left\langle a b^{-1}\right\rangle$ travels by $a .$
A finite group is called Hamiltonian if all of its subgroups are normal. (One non-Abelian example is $Q_{4}$.) Show that Theorem $30.3$ can be generalized to include all Hamiltonian groups.
(Factor Group Lemma) Let $S$ be a generating set for a group $G$, let $N$ be a cyclic normal subgroup of $G$, and let$$\bar{S}=\{s N \mid s \in S\}$$If $\left(a_{1} N, \ldots, a_{r} N\right)$ is a Hamiltonian circuit in $\operatorname{Cay}(\bar{S}: G / N)$ and the product $a_{1} \cdots a_{r}$ generates $N$, prove that$$|N| *\left(a_{1}, \ldots, a_{r}\right)$$is a Hamiltonian circuit in $\operatorname{Cay}(S: G)$.