The problem of partitioning a large complex network equitably into sparse modules with given rules can be modeled by the equitable list d-degenerate coloring of graphs. This paper establishes theoretical results on such a coloring based on a newly proposed conjecture which states that every graph G is equitably d-degenerate k-colorable and equitably d-degenerate k-choosable for every integer $k\ge (\mathrm{\Delta }(G)+1)/(d+1)$. This conjecture is strong as it implies the Hajnal-Szemerédi theorem on equitable coloring, the equitable list coloring conjecture (Kostochka, Pelsmajer, and West, 2003), the equitable vertex arboricity conjecture (Wu, Zhang, and Li, 2013), and the equitable list vertex arboricity conjecture (Zhang, 2016). In this paper, we confirm this unified conjecture for globally coupled networks, $(d+1)$-degenerate graphs, graphs with bounded maximum average degree, and planar graphs with large maximum degree. The equitable d-degenerate k-colorability part of this conjecture is also verified for interval graphs, generalizing a result of Niu, Li, and Zhang (2021).

用给定规则将大型复杂网络公平地划分为稀疏模块的问题可以通过图的等价列表d-简并着色来建模。本文建立在这样一个理论结果着色基于新提出的猜想其中指出每个图形ģ是公平d -degenerate ķ -colorable和公平d -degenerate ķ -choosable每整数$ķ\ge （\mathrm{\Delta }（G）+\mathrm{1个}）/（d+\mathrm{1个}）$。该猜想很强，因为它暗示了关于均等着色的Hajnal-Szemerédi定理，等价列表着色猜想（Kostochka，Pelsmajer和West，2003年），等价顶点任意度猜想（Wu，Zhang和Li，2013年）以及公平列表顶点任意性猜想（Zhang，2016）。在本文中，我们确认了针对全球耦合网络的统一猜想，$（d+\mathrm{1个}）$-退化图，有界最大平均度的图和平面图的最大度数大。该猜想的等价d-简并k-色性部分也得到了区间图的验证，从而概括了Niu，Li和Zhang（2021）的结果。