Chromatic puzzle

Chromatic it's relax puzzle game without timer.
You need to color all vertices. That's all! But ... neighboring vertices cannot have the same color.
So, the aim is to use the fewest number of colors, and the chromatic number of a graph is the smallest number of colors that suffice for a coloring.
You will begin with vertex coloring, where one colors the vertices of a graph in such a way that adjacent vertices get different colors. You will start with some easy examples, and then move on to more complicated graphs. Sometimes it's easy - sometimes it's very very difficult.

A little bit of mathematics history:
The chromatic number of a graph is the smallest number of colors needed to color the vertices of graph so that no two adjacent vertices share the same color.
The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. Guthrie’s brother passed on the question to his mathematics teacher Augustus de Morgan at University College, who mentioned it in a letter to William Hamilton in 1852. Arthur Cayley raised the problem at a meeting of the London Mathematical Society in 1879. The same year, Alfred Kempe published a paper that claimed to establish the result, and for a decade the four color problem was considered solved. For his accomplishment Kempe was elected a Fellow of the Royal Society and later President of the London Mathematical Society.