Extremal Theta-free planar graphs

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Abstract

Given a family F, a graph is F-free if it does not contain any graph in F as a subgraph. We continue to study the topic of “extremal” planar graphs initiated by Dowden (2016), that is, how many edges can an F-free planar graph on n vertices have? We define exP(n,F) to be the maximum number of edges in an F-free planar graph on n vertices. Dowden obtained the tight bounds exP(n,C4)15(n2)7 for all n4 and exP(n,C5)(12n33)5 for all n11. In this paper, we continue to promote the idea of determining exP(n,F) for certain classes F. Let Θk denote the family of Theta graphs on k4 vertices, that is, graphs obtained from a cycle Ck by adding an additional edge joining two non-consecutive vertices. The study of exP(n,Θ4) was suggested by Dowden. We show that exP(n,Θ4)12(n2)5 for all n4, exP(n,Θ5)5(n2)2 for all n5, and then demonstrate that these bounds are tight, in the sense that there are infinitely many values of n for which they are attained exactly. We also prove that exP(n,C6)exP(n,Θ6)18(n2)7 for all n6.

Keywords

Turán number
Extremal graph
Planar graph

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