DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvořak and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on sufficient conditions for planar graphs to be DP-$4$-colorable. In particular, for each $k \in \{3, 4, 5, 6\}$, every planar graph without $k$-cycles is DP-$4$-colorable. In this paper, we prove that every planar graph without $7$-cycles and butterflies is DP-$4$-colorable. Our proof can be easily modified to prove other sufficient conditions that forbid clusters formed by many triangles.

中文翻译：

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没有 7 循环和蝴蝶的平面图是 DP-4 可着色的

DP-coloring（也称为对应着色）是列表着色的泛化，由 Dvořak 和 Postle 在 2017 年提出。众所周知，存在非 4-choosable 平面图。最近很多注意力都放在平面图是 DP-$4$-可着色的充分条件上。特别地，对于每个 $k \in \{3, 4, 5, 6\}$，每个没有 $k$-cycles 的平面图都是 DP-$4$-可着色的。在本文中，我们证明了每个没有 $7$-cycles 和蝴蝶的平面图都是 DP-$4$-可着色的。我们的证明可以很容易地修改以证明其他充分条件，禁止由许多三角形形成集群。