K-Cyclic chromatic number

[math]\displaystyle{ k }[/math]-Cyclic chromatic number --- [math]\displaystyle{ k }[/math]-циклическое хроматическое число.

The [math]\displaystyle{ k }[/math]-cyclic chromatic number [math]\displaystyle{ \chi_{k}(G) }[/math] of a plane graph is the smallest number of colours in a vertex colouring of [math]\displaystyle{ G }[/math] such that no face of size at most [math]\displaystyle{ k }[/math] has two boundary vertices of the same colour. It is easy to see that the Four Colour Theorem may be stated in the form:

[math]\displaystyle{ \chi_{3}(G) \leq 4 }[/math]

for every plane graph [math]\displaystyle{ G }[/math].

The number [math]\displaystyle{ \chi_{k}(G) }[/math] was introduced explicitly by Ore and Plummer (1969).