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Sharp linear Turán remainders for cycle and edge extremality
Abstract
For a graph G, let e(G) denote the number of edges of G, and let c(G) be the number of distinct cycles in G. Morrison, Roberts and Scott asked whether, for every fixed graph H and all large n, some n-vertex H-free graph maximizes both e(G) and c(G). In this paper, we show that the answer is no for every possible chromatic number. Let Tn,r be the complete r-partite Turán graph on n vertices. We find a constant γr>0 such that if n is sufficiently large and ℋ is a finite family with χ(ℋ)=r+1 satisfying ex(n,ℋ)<e(Tn,r)+γr n, then every cycle-maximal ℋ-free graph is edge-extremal, and γr is best possible.
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BibTeX
@misc{lin2026sharp,
title = {{Sharp linear Turán remainders for cycle and edge extremality}},
author = {Jifu Lin and Xiaolin Wang and Guangmiao Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12109},
eprint = {2610.12109},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12109},
}
RIS
TY - UNPB
TI - Sharp linear Turán remainders for cycle and edge extremality
AU - Jifu Lin
AU - Xiaolin Wang
AU - Guangmiao Yu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12109
UR - https://arxiv.org/abs/2610.12109
AB - For a graph G, let e(G) denote the number of edges of G, and let c(G) be the number of distinct cycles in G. Morrison, Roberts and Scott asked whether, for every fixed graph H and all large n, some n-vertex H-free graph maximizes both e(G) and c(G). In this paper, we show that the answer is no for every possible chromatic number. Let T_n,r be the complete r-partite Turán graph on n vertices. We find a constant γ_r>0 such that if n is sufficiently large and ℋ is a finite family with χ(ℋ)=r+1 satisfying ex(n,ℋ)<e(T_n,r)+γ_r n, then every cycle-maximal ℋ-free graph is edge-extremal, and γ_r is best possible.
ER -