Short Cycles in Planar Graphs

Abstract

We present new algorithms for finding short cycles (of length at most 6) in planar graphs. Although there is an O(n) algorithm for finding any fixed subgraph H in a given n-vertex planar graph [5], the multiplication constant hidden in “O” notation (which depends on the size of H) is so high, that it rather cannot be used in practice even when |V(H)| = 4. Our approach gives faster “practical algorithms” which are additionally much easier to implement.

As a side-effect of our approach we show that the maximum number of k-cycles in n-vertex planar graph is $\Theta (n^{\lfloor \frac{k}{2} \rfloor})$.