A graph is even (resp. odd) if all its vertex degrees are even (resp. odd). We consider edge coverings by prescribed number of even and/or odd subgraphs. In view of the 8-Flow Theorem, a graph admits a covering by three even subgraphs if and only if it is bridgeless. Coverability by three odd subgraphs has been characterized recently [ Petruševski, M.; Škrekovski, R. Coverability of graph by three odd subgraphs. J. Graph Theory2019, 92, 304–321]. It is not hard to argue that every acyclic graph can be decomposed into two odd subgraphs, which implies that every graph admits a decomposition into two odd subgraphs and one even subgraph. Here, we prove that every 3-edge-connected graph is coverable by two even subgraphs and one odd subgraph. The result is sharp in terms of edge-connectivity. We also discuss coverability by more than three parity regular subgraphs, and prove that it can be efficiently decided whether a given instance of such covering exists. Moreover, we deduce here a polynomial time algorithm which determines whether a given set of edges extends to an odd subgraph. Finally, we share some thoughts on coverability by two subgraphs and conclude with two conjectures.

中文翻译：

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奇偶校验子图对图的可覆盖性

如果图形的所有顶点度均为偶数（奇数），则该图为偶数（奇数）。我们按规定数量的偶数和/或奇数子图考虑边缘覆盖。根据8流定理，当且仅当图是无桥的时，图才允许覆盖三个偶数子图。最近已经描述了三个奇数子图的可覆盖性[Petruševski，M .; Škrekovski，R.被三个奇数子图覆盖的图。J.图论2019，92，304–321]。不难争论每个非循环图都可以分解为两个奇数子图，这意味着每个图都允许分解为两个奇数子图和一个偶数子图。在这里，我们证明了每个三边连接图都可以被两个偶数子图和一个奇数子图覆盖。就边缘连接性而言，结果很清晰。我们还将通过三个以上的奇偶正则子图讨论可覆盖性，并证明可以有效地确定是否存在此类覆盖的给定实例。此外，我们在这里推导多项式时间算法，该算法确定给定的一组边是否扩展到奇数子图。最后，我们通过两个子图分享​​了关于可覆盖性的一些想法，并得出了两个推测。