Outerplanar Tur\'an number of a cycle
Ervin Gy\H{o}ri, Guilherme Zeus Dantas e Moura, Runtian Zhou

TL;DR
This paper provides a shorter proof for the maximum number of edges in an outerplanar graph with n vertices that avoids a cycle of length k, refining the understanding of extremal graph properties.
Contribution
The paper introduces a dual graph technique to establish a sharper upper bound for the outerplanar Turán number of cycles, simplifying previous proofs.
Findings
Established a shorter proof for the upper bound of $ex_ ext{OP}(n,C_k)$
Derived a precise upper bound formula for outerplanar Turán numbers of cycles
Enhanced the methodological approach using dual graph techniques
Abstract
A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let be a graph. The outerplanar Tur\'an number of , denoted by , is the maximum number of edges in an -vertex outerplanar graph which does not contain as a subgraph. In 2021, L. Fang et al. determined the outerplanar Tur\'an number of cycles and paths. In this paper, we use techniques of dual graph to give a shorter proof for the sharp upperbound of .
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research · Limits and Structures in Graph Theory
