Planar Turán number of the disjoint union of cycles.

The planar Turán number of H , denoted by e x P (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 3 n − 6. Lan, Shi and Song determine the exact value of e x P (n , 2 C 3). We continue to study planar Turán number of two vertex-disjoint union of cycles and obtain the exact value of e x P (n , H) , where H is vertex-disjoint union of C 3 and C 4. The extremal graphs are also characterized. We also improve the lower bound of e x P (n , 2 C k) when n is sufficiently large. [ABSTRACT FROM AUTHOR]