Exact planar Turán numbers are established for K1+(P2∪P3), a combined C3/Θ4 configuration, and the disjoint union of C3 and Θ4, with extremal graph characterizations.
Dense $2$-connected planar graphs and the planar Tur\'{a}n number of $2C_k$
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abstract
Shi, Walsh and Yu demonstrated that any dense planar graph with certain property (known as circuit graph) contains a large near-triangulation. We extend the result to $2$-connected plane graphs, thereby addressing a question posed by them. Using the result, we prove that the planar Tu\'{a}n number of $2C_k$ is $\left[3-\Theta(k^{\log_23})^{-1}\right]n$ when $k\geq 5$.
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Planar Tur\'{a}n numbers of three configurations
Exact planar Turán numbers are established for K1+(P2∪P3), a combined C3/Θ4 configuration, and the disjoint union of C3 and Θ4, with extremal graph characterizations.