We use $N^+$ to denote the set of positive integers. For any finite array $A:(a\_1,b\_1),...,(a\_k,b\_k)$, where every $(a\_i,b\_i)\in N^+\times N^+$, we call $A$ is good if for every $i\in \{a\_1,b\_1,...,a\_k,b\_k\}$, $i$ appears exactly two times in $a\_1,b\_1,...,a\_k,b\_k$. For example, $A\_1:(1,1)$,$A\_2:(1,2),(1,2)$ and $A\_3:(1,2),(2,1)$ are all good, but $A\_4:(1,2),(1,3)$ is not good. Moreover we call a good array $A$ is compact if there does not exist a proper sub-array of $A$ which is good. For example,$A:(1,2),(3,4),(2,1),(4,3)$ is good but not compact, because $(1,2),(2,1)$ is a proper sub-array of $A$ which is also good.


We define a good $N^+$-cycle $C=a\_1$-$a\_2$-...-$a\_n$-$a\_1$ is a "diamond necklace" such that every "diamond" $a\_i$ is a positive integer and appears exactly two times in $a\_1,a\_2,...,a\_n$. For example,$1$-$2$-$3$-$1$-$2$-$3$-$1$ is a good $N^+$-cycle. I conjecture that for every $N^+$-cycle $C=a\_1$-$a\_2$-...-$a\_n$-$a\_1$, there must exist $1\leq i\_1<i\_2<...i\_k\leq n$ such that


$(1）i\_2-i\_1\geq 2,...,i\_k-i\_{k-1}\geq 2,i\_1+n-i\_k\geq 2$;


$(2)(a\_{i\_1},a\_{i\_2-1}),(a\_{i\_2},a\_{i\_3-1}）...,(a\_{i\_{k-1}},a\_{i\_k-1}),(a\_{i\_k},a\_{i\_1-1})$$($when $i\_1=1$,let $a\_{i\_1-1}=a\_0=a\_n$$)$ is a compact good array.


$($Remark: just like cut the "diamond necklace" into $k$ paragraphs: $(a\_{i\_1}$-$a\_{i\_1+1}$-...-$a\_{i\_2-1})$-$(a\_{i\_2}$-$a\_{i\_2+1}$-...-$a\_{i\_3-1})$-...-$(a\_{i\_{k-1}}$-$a\_{i\_{k-1}+1}$-...-$a\_{i\_k-1})$-$(a\_{i\_k}$-$a\_{i\_k+1}$-...-$a\_{i\_1-1})$-$)$


Is it true?



The "compact good array" in a "good $N^+$-cycle"
The conjecture is right. It is equivalent to the claim below:


Let G be a simple graph which is a $2n$-cycle equipped with $n$ chords such that $G$ is $3$-regular, in other words, the set of the $n$ chords is a perfect matching of $G$ (that is, every vertex of $G$ is matched). Then there must exist at least two different $2n$-cycles in $G$.


The proof is given by Tony Huynh, see my other question: [Does this graph contain at least two Hamiltonian cycles?](https://mathoverflow.net/questions/145531/an-interesting-conjecture-about-even-cycle).


