European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2021-03-02 , DOI: 10.1016/j.ejc.2021.103320 Maria Axenovich , David Offner , Casey Tompkins
We consider edge decompositions of the -dimensional hypercube into isomorphic copies of a given graph . While a number of results are known about decomposing into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if is even, and divides the number of edges of , then the path of length decomposes . Tapadia et al. proved that any path of length , where , satisfying these conditions decomposes . Here, we make progress toward resolving Erde’s conjecture by showing that cycles of certain lengths up to decompose . As a consequence, we show that can be decomposed into copies of any path of length at most dividing the number of edges of , thereby settling Erde’s conjecture up to a linear factor.
中文翻译:
甚至超立方体的长路径和循环分解
我们考虑了边的分解 维超立方体 变成给定图的同构副本 。虽然有许多关于分解的结果在各种类别的图中,给定长度的路径和循环的最简单情况还远远没有被理解。埃尔德(Erde)的一个猜想断言,如果 甚至, 和 划分的边数 ,然后是长度的路径 分解 。Tapadia等。证明任何长度的路径, 在哪里 ,满足这些条件分解 。在这里,我们通过显示一定长度的周期直到 分解 。结果,我们证明了 最多可以分解为任何长度的路径的副本 除以的边数 ,从而使Erde猜想的线性度最高。