Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on them with special structures. In this series of two papers, we characterize entropy functions on the 2-dimensional faces of the polymatroidal region of degree 4. In Part I, we formulate the problem, enumerate all 59 types of 2-dimensional faces of the region by an algorithm, and fully characterize entropy functions on 49 types of them. Among them, those non-trivial cases are mainly characterized by the graph-coloring technique. The entropy functions on the remaining 10 types of faces will be characterized in Part II, among which 8 types are fully characterized, and 2 types are partially characterized.

翻译：熵函数的刻画在信息论中具有基础性重要意义。通过对香农外边界（即多拟阵区域）施加约束，可获得该区域的各维面及其上具有特殊结构的熵函数。在本系列两篇论文中，我们刻画了四维多拟阵区域二维面上的熵函数。在第一部分中，我们构建了问题框架，通过算法枚举了该区域全部59类二维面，并完整刻画了其中49类面上的熵函数。其中非平凡情形主要借助图着色技术进行刻画。剩余10类面上的熵函数将在第二部分中完成刻画，其中8类已获完整刻画，2类获得部分刻画。