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The Exact Number of Pentagons in Book-Free Graphs
Abstract
Let Bk denote the book consisting of k triangles sharing a common edge. We prove that there exist absolute constants ρ>0 and N such that, for every pair of integers n,k with n≥ N and 2≤ k≤ ρn, writing n=5q+r with 0≤ r<5, ex(n,C5,Bk)=q 5−r(q+1)r, with the balanced complete C5-blow-ups as the only extremal graphs. The linear order of this range is best possible since for all sufficiently large n, the formula fails whenever k≥0.14n.
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BibTeX
@misc{hu2026exact,
title = {{The Exact Number of Pentagons in Book-Free Graphs}},
author = {Wentao Hu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06753},
eprint = {2611.06753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06753},
}
RIS
TY - UNPB
TI - The Exact Number of Pentagons in Book-Free Graphs
AU - Wentao Hu
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06753
UR - https://arxiv.org/abs/2611.06753
AB - Let B_k denote the book consisting of k triangles sharing a common edge. We prove that there exist absolute constants ρ>0 and N such that, for every pair of integers n,k with n≥ N and 2≤ k≤ ρn, writing n=5q+r with 0≤ r<5, ex(n,C_5,B_k)=q^ 5−r(q+1)^r, with the balanced complete C_5-blow-ups as the only extremal graphs. The linear order of this range is best possible since for all sufficiently large n, the formula fails whenever k≥0.14n.
ER -