| In this paper, we study the upper limit of the shortest cycles in2-connected simple planar graph G of order n. Suppose the cycle length of the shortest cycle C is k, discuss the possible values of k and then we can determine the upper limit. For different value of k, we consider:(1) When k=2i+1,i≥1, we use breadth first search and the properties of the fundamental cycle.(2) When k=4, discuss the shortest cycles intersection and then use a special4cycle.(3) When k=2i, i≥3, this situation is similar to the situation (1) and use Jordan Curve Theorem. Finally, we show that the number of the shortest cycles in2-connected simple planar graph G of order n is O(n2). The graph Kn-2,2shows that the result is the best. |
