A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph G into k locally irregular subgraphs is a partition \(E_1,\dots ,E_k\) of E(G) into k parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit such a decomposition into at most three locally irregular subgraphs. This conjecture was verified for a few graph classes in recent years. This work is dedicated to the decomposability of degenerate graphs with low degeneracy. Our main result is that decomposable k-degenerate graphs decompose into at most \(3k+1\) locally irregular subgraphs, which improves on previous results whenever \(k \le 9\). We improve this result further for some specific classes of degenerate graphs, such as bipartite cacti, k-trees, and planar graphs. Although our results provide only little progress towards the leading conjecture above, the main contribution of this work is rather the decomposition schemes and methods we introduce to prove these results.

如果一个（无向）图的相邻顶点中没有两个顶点的度数相同，则它是局部不规则的。的曲线图的分解ģ成ķ局部不规则的子图是分区\（E_1，\点，E_k \）的È（ģ）插入ķ份其中的每一个诱导局部不规则子图。并非所有图都分解为局部不规则子图；但是，可以推测，只要有图，它就应该允许将这种分解分解为最多三个局部不规则子图。近年来，对于一些图类都对此猜想进行了验证。这项工作致力于具有低退化性的退化图的可分解性。我们的主要结果是可分解的k-简并图最多分解为\（3k + 1 \）个局部不规则子图，每当\（k \ le 9 \）时，该结果就会得到改善。对于某些特定类别的简并图，例如二分仙人掌，k树和平面图，我们将进一步改善此结果。尽管我们的结果在上述领先猜想上进展甚微，但这项工作的主要贡献在于我们引入了证明这些结果的分解方案和方法。