On the average crosscap number Ⅱ: Bounds for a graph

摘要

The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less than (2~β(G)-1)/(2~β(G)-1) and not larger than β(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.