THE PLANAR TURÁN NUMBER OF {C6, C7}

Let H be a set of graphs. A graph is H-free if it does not contain any copy of H as a subgraph where H is an element of H. The planar Turan number of H, denoted by exp(n, H), is the maximum number of edges in an H-free planar graph on n vertices. The upper bounds of exp(n, {C-k,C- Ck+1}) are known when 3 < k < 5, and these bounds are tight. In this paper, we give the upper bound of exp(n, {C6, C7}) for all integers n >= 76, and this bound is sharp.