In the problem of 2-coloring without monochromatic triangles (or triangle-tree 2-coloring), vertices of the simple, connected, undirected graph are colored with either 'black' or 'white' such that there are no 3 mutually adjacent vertices of the same color. In this paper we are positively answering the question posed in our previous work, namely, if there exists an algorithm solving 2-coloring without monochromatic triangles on planar graphs with linear-time complexity.

翻译：在没有单色三角形(或三角树2色)的2色色化问题中,简单、相连、无方向的图形的顶端颜色为“黑色”或“白色”,因此没有3个相邻的相同颜色的顶端。在本文中,我们正在积极回答我们先前工作中提出的问题,即,如果在具有线性复杂度的平面图上存在一种算法来解决2色化,而没有单色三角形。