Arithmetic progressions of cycles in outer-planar graphs Original Research Article

A question of Erdős asks if every graph with minimum degree 3 must contain a pair of cycles whose lengths differ by 1 or 2. Some recent work of Häggkvist and Scott (see Arithmetic progressions of cycles in graphs, preprint), whilst proving this, also shows that minimum degree 500k2 guarantees the existence of cycles whose lengths are m,m+2,m+4,…,m+2k for some m—an arithmetic progression of cycles. In like vein, we prove that an outer-planar graph of order n, with bounded internal face size, and outer face a cycle, must contain a sequence of cycles whose lengths form an arithmetic progression of length exp((c log n)1/3−log log n). Using this we give an answer for outer-planar graphs to a question of Erdős concerning the number of different sets which can be achieved as cycle spectra.