(k,g)-Cage

[math]\displaystyle{ \,(k,g) }[/math]-Cage — [math]\displaystyle{ \,(k,g) }[/math]-клетка.

For a given ordered pair of integers [math]\displaystyle{ \,(k,g) }[/math], with [math]\displaystyle{ k \geq 2 }[/math] and [math]\displaystyle{ g \geq 3 }[/math], a [math]\displaystyle{ \,k }[/math]-regular graph with the smallest cycle length, or girth, equal to [math]\displaystyle{ \,g }[/math] is said to be a [math]\displaystyle{ \,(k,g) }[/math]-graph. A [math]\displaystyle{ \,(k,g) }[/math]-cage is a [math]\displaystyle{ \,(k,g) }[/math]-graph having the least number, [math]\displaystyle{ \,f(k,g) }[/math], of vertices. We call [math]\displaystyle{ \,f(k,g) }[/math] the cage number of a [math]\displaystyle{ \,(k,g) }[/math]-graph. One readily observes that [math]\displaystyle{ \,(2,g) }[/math]-cages are cycles of length [math]\displaystyle{ \,g }[/math], and [math]\displaystyle{ \,(k,3) }[/math]-cages are complete graphs of order [math]\displaystyle{ \,k+1 }[/math].

The unique [math]\displaystyle{ \,(3,7) }[/math]-cage known as the McGee graph is an example of a cage that is not transitive. It has [math]\displaystyle{ \,24 }[/math] vertices and its automorphism group has order [math]\displaystyle{ \,32 }[/math].

Литература

• Евстигнеев В.А., Касьянов В.Н. Словарь по графам в информатике. — Новосибирск: Сибирское Научное Издательство, 2009.