Planar Turán number of the disjoint union of cycles

Abstract

The planar Turán number of H, denoted by exP(n,H), is the maximum number of edges in an n-vertex H-free planar graph. The planar Turán number of k≥3 vertex-disjoint union of cycles is a trivial value 3n−6. Lan, Shi and Song determine the exact value of exP(n,2C3). We continue to study planar Turán number of two vertex-disjoint union of cycles and obtain the exact value of exP(n,H), where H is vertex-disjoint union of C3 and C4. The extremal graphs are also characterized. We also improve the lower bound of exP(n,2Ck) when n is sufficiently large.