Planar Turan number of the disjoint union of cycles

The planar Turan number of H, denoted by ex(P)(n, H), is the maximum number of edges in an n-vertex H-free planar graph. The planar Turan number of k >= 3 vertex-disjoint union of cycles is a trivial value 3n - 6. Lan, Shi and Song determine the exact value of ex(P)(n, 2C(3)). We continue to study planar Turan number of two vertex-disjoint union of cycles and obtain the exact value of ex(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 ex(P)(n, 2C(k)) when n is sufficiently large.(c) 2023 Published by Elsevier B.V.