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An entropy proof of the commonality of apices of trees

  • Hao Chen
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Abstract

A graph H is called common if the number of monochromatic copies of H in a $2$-edge-colouring of a large complete graph is asymptotically minimized by the random colouring with an equal proportion of each colour. Grzesik, Lee, Lidický and Volec conjectured that every k-apex of any connected Sidorenko graph is common. Recently, Kr{á}l', Krnc and Lamaison proved that the k-apex of any tree is common for k≤ 5. In this paper, we extend their result by showing that for every positive integer k, the k-apex of any tree or any forest with average degree at least one is common. Our proof involves an application of the entropy method.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026entropy,
  title = {{An entropy proof of the commonality of apices of trees}},
  author = {Hao Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29469},
  eprint = {2610.29469},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29469},
}
RIS
TY  - UNPB
TI  - An entropy proof of the commonality of apices of trees
AU  - Hao Chen
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29469
UR  - https://arxiv.org/abs/2610.29469
AB  - A graph H is called common if the number of monochromatic copies of H in a $2$-edge-colouring of a large complete graph is asymptotically minimized by the random colouring with an equal proportion of each colour. Grzesik, Lee, Lidický and Volec conjectured that every k-apex of any connected Sidorenko graph is common. Recently, Kr{á}l', Krnc and Lamaison proved that the k-apex of any tree is common for k≤ 5. In this paper, we extend their result by showing that for every positive integer k, the k-apex of any tree or any forest with average degree at least one is common. Our proof involves an application of the entropy method.
ER  -