Sharp planar Turán bounds for quasi-double stars
Abstract
We study W-free planar graphs for W∈{W2,4,W2,5,W3,4}, where the quasi-double star Wh,k is obtained from a three-vertex path by attaching h leaves to one endpoint and k leaves to the other. We prove that every W2,4-free planar graph on n vertices has at most 9n/4 edges, and the bound is attained whenever 8| n. This determines the planar Turán density of W2,4 as 9/4. We also establish the sharp upper bound 5n/2 for W2,5. Combined with known constructions of planar graphs of maximum degree five, it yields exPP(n,W2,5)=⌊5n/2⌋ for every n≥15. These results close the two corresponding coefficient gaps in the bounds of Liu et~al. Our proofs use structural restrictions on high-degree vertices, local deletions, and degree deficits in neighborhoods of radius two. For W3,4, we characterize the planar graphs with a dominating vertex that avoid this tree and determine their exact extremal number, ⌊(5n−7)/2⌋, for every n≥10. Finally, W-free planar triangulations have at most eight, twelve, and eleven vertices for W=W2,4,W2,5,W3,4, respectively; the first two bounds are sharp.
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BibTeX
@misc{shao2026sharp,
title = {{Sharp planar Turán bounds for quasi-double stars}},
author = {Zehui Shao and Enqiang Zhu and Shaohui Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05706},
eprint = {2610.05706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05706},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp planar Turán bounds for quasi-double stars
AU - Zehui Shao
AU - Enqiang Zhu
AU - Shaohui Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05706
UR - https://arxiv.org/abs/2610.05706
AB - We study W-free planar graphs for W∈{W_2,4,W_2,5,W_3,4}, where the quasi-double star W_h,k is obtained from a three-vertex path by attaching h leaves to one endpoint and k leaves to the other. We prove that every W_2,4-free planar graph on n vertices has at most 9n/4 edges, and the bound is attained whenever 8| n. This determines the planar Turán density of W_2,4 as 9/4. We also establish the sharp upper bound 5n/2 for W_2,5. Combined with known constructions of planar graphs of maximum degree five, it yields ex_PP(n,W_2,5)=⌊5n/2⌋ for every n≥15. These results close the two corresponding coefficient gaps in the bounds of Liu et~al. Our proofs use structural restrictions on high-degree vertices, local deletions, and degree deficits in neighborhoods of radius two. For W_3,4, we characterize the planar graphs with a dominating vertex that avoid this tree and determine their exact extremal number, ⌊(5n−7)/2⌋, for every n≥10. Finally, W-free planar triangulations have at most eight, twelve, and eleven vertices for W=W_2,4,W_2,5,W_3,4, respectively; the first two bounds are sharp.
ER -