论文标题
高维的反示例
Higher-dimensional counterexamples to Hamiltonicity
论文作者
论文摘要
对于$ d \ ge 2 $，我们显示所有$ d $ - polytopes的图形都具有汉密尔顿线图，并且仅当$ d \ ne 3 $：我们展示了$ 252 $ pertices上的$ 3 $ polytope的图表，其线路图甚至没有汉密尔顿路径。适应Grünbaum和Motzkin的建筑，对于大$ n $，我们还构建了简单的$ 3 $ - polytopes，上面$ 3N $的顶点，其line Gragr Any Simple Path的任何简单路径都短于$ 10 n^α$，对于某种常数$α<1 $。此外，我们为汉密尔顿图理论中四个著名结果的简单复合物提供了四个基本反示例。
For $d \ge 2$, we show that all graphs of $d$-polytopes have a Hamiltonian line graph if and only if $d \ne 3$: We exhibit a graph of a $3$-polytope on $252$ vertices whose line graph does not even have Hamiltonian paths. Adapting a construction by Grünbaum and Motzkin, for large $n$ we also construct simple $3$-polytopes on $3n$ vertices in whose line graph any simple path is shorter than $10 n^α$, for some constant $α<1$. Moreover, we give four elementary counterexamples of plausible extensions to simplicial complexes of four famous results in Hamiltonian graph theory.
