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The edge spectral extremal problem for kK3 in nonzero residue classes

  • Jing Gao
  • Shuchao Li
  • Yuantian Yu
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Abstract

For a fixed integer k≥ 2, let kK3 denote the vertex-disjoint union of k triangles. A recent fixed-size spectral theorem of Das and Yamini asserts that, for all sufficiently large m, every kK3-free graph G of size m satisfies λ(G)≤ (k−1)+√(m−k(k−1)), and equality holds if and only if (2k−1)| m and $ G\cong \bigl(K_{2k-1}\vee qK_1\bigr)\cup tK_1,\, q=\frac{m}{2k-1}-(k-1) $ for some t≥ 0. They explicitly posed the open problem: Let k≥ 2 be fixed and ℓ be a residue in {1,…,2k−2}. For all sufficiently large integers m≡ ℓpmod2k−1, determine the exact value of $\max\bigl\{λ(G): e(G)=m,\ G\text{ is }kK_3\text{-free}\bigr\}, $ and characterize all graphs attaining this maximum. In this paper, using the positive-defect version of the bounded-core method for divisible sizes together with several new ideas developed in this paper, we give a complete solution to the aforementioned open problem.

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Cite

BibTeXRIS
BibTeX
@misc{gao2026edge,
  title = {{The edge spectral extremal problem for $kK_3$ in nonzero residue classes}},
  author = {Jing Gao and Shuchao Li and Yuantian Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07022},
  eprint = {2610.07022},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07022},
}
RIS
TY  - UNPB
TI  - The edge spectral extremal problem for kK_3 in nonzero residue classes
AU  - Jing Gao
AU  - Shuchao Li
AU  - Yuantian Yu
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.07022
UR  - https://arxiv.org/abs/2610.07022
AB  - For a fixed integer k≥ 2, let kK_3 denote the vertex-disjoint union of k triangles. A recent fixed-size spectral theorem of Das and Yamini asserts that, for all sufficiently large m, every kK_3-free graph G of size m satisfies λ(G)≤ (k−1)+√(m−k(k−1)), and equality holds if and only if (2k−1)| m and $ G\cong \bigl(K_{2k-1}\vee qK_1\bigr)\cup tK_1,\, q=\frac{m}{2k-1}-(k-1) $ for some t≥ 0. They explicitly posed the open problem: Let k≥ 2 be fixed and ℓ be a residue in {1,…,2k−2}. For all sufficiently large integers m≡ ℓpmod2k−1, determine the exact value of $\max\bigl\{λ(G): e(G)=m,\ G\text{ is }kK_3\text{-free}\bigr\}, $ and characterize all graphs attaining this maximum. In this paper, using the positive-defect version of the bounded-core method for divisible sizes together with several new ideas developed in this paper, we give a complete solution to the aforementioned open problem.
ER  -