论文标题
在将笛卡尔产品顶点逆转导向的边缘之后
Hamiltonicity after reversing the directed edges at a vertex of a Cartesian product
论文作者
论文摘要
令$ \ vec c_m $和$ \ vec c_n $是指导长度$ m $和$ n $的周期，带有$ m，n \ ge 3 $，然后让$ p（\ vec c_m \ vec c_m \ vec mathbin {\ box} \ box} \ vec c_n）$是从Cartesian Focuct $ \ vec \ vec c_的digraph \ vec c_n $通过选择顶点$ v $，并逆转所有四个有针对$ v $的有向边的方向。 （此操作称为“推动”在顶点$ v $。）通过应用S.X.WU未发表的特殊情况，我们发现基本的数字理论理论是在$ p（\ vec c_m \ mathbin {\ box} \ box} \ vec c_n）$ p（\ vec c_m \ vec c_m \ vec c_m \ vec c_m \ vec c_n）中的必要条件和足够条件。 结果是，如果$ p（\ vec c_m \ mathbin {\ box} \ vec c_n）$是hamiltonian，则$ \ gcd（m，n）= 1 $，这意味着$ \ vec c_m \ vec c_m \ mathbin {\ box} {\ box} \ vec c_n $不是Hamiltonian。最终结论验证了J.B. Klerlein和E.C. Carr的猜想。
Let $\vec C_m$ and $\vec C_n$ be directed cycles of length $m$ and $n$, with $m,n \ge 3$, and let $P(\vec C_m \mathbin{\Box} \vec C_n)$ be the digraph that is obtained from the Cartesian product $\vec C_m \mathbin{\Box} \vec C_n$ by choosing a vertex $v$, and reversing the orientation of all four directed edges that are incident with $v$. (This operation is called "pushing" at the vertex $v$.) By applying a special case of unpublished work of S.X.Wu, we find elementary number-theoretic necessary and sufficient conditions for the existence of a hamiltonian cycle in $P(\vec C_m \mathbin{\Box} \vec C_n)$. A consequence is that if $P(\vec C_m \mathbin{\Box} \vec C_n)$ is hamiltonian, then $\gcd(m,n) = 1$, which implies that $\vec C_m \mathbin{\Box} \vec C_n$ is not hamiltonian. This final conclusion verifies a conjecture of J.B.Klerlein and E.C.Carr.
