On almost Gallai colourings in complete graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916660305199104 |
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| author | Grebennikov, Alexandr Mattos, Letícia Szabó, Tibor |
| author_facet | Grebennikov, Alexandr Mattos, Letícia Szabó, Tibor |
| contents | For $t \in \mathbb{N}$, we say that a colouring of $E(K_n)$ is $\textit{almost}$ $t$-$\textit{Gallai}$ if no two rainbow $t$-cliques share an edge. Motivated by a lemma of Berkowitz on bounding the modulus of the characteristic function of clique counts in random graphs, we study the maximum number $τ_t(n)$ of rainbow $t$-cliques in an almost $t$-Gallai colouring of $E(K_n)$. For every $t \ge 4$, we show that $n^{2-o(1)} \leq τ_t(n) = o(n^2)$. For $t=3$, surprisingly, the behaviour is substantially different. Our main result establishes that $$\left ( \frac{1}{2}-o(1) \right ) n\log n \le τ_3(n) = O\big (n^{\sqrt{2}}\log n \big ),$$ which gives the first non-trivial improvements over the simple lower and upper bounds. Our proof combines various applications of the probabilistic method and a generalisation of the edge-isoperimetric inequality for the hypercube. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_17334 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On almost Gallai colourings in complete graphs Grebennikov, Alexandr Mattos, Letícia Szabó, Tibor Combinatorics For $t \in \mathbb{N}$, we say that a colouring of $E(K_n)$ is $\textit{almost}$ $t$-$\textit{Gallai}$ if no two rainbow $t$-cliques share an edge. Motivated by a lemma of Berkowitz on bounding the modulus of the characteristic function of clique counts in random graphs, we study the maximum number $τ_t(n)$ of rainbow $t$-cliques in an almost $t$-Gallai colouring of $E(K_n)$. For every $t \ge 4$, we show that $n^{2-o(1)} \leq τ_t(n) = o(n^2)$. For $t=3$, surprisingly, the behaviour is substantially different. Our main result establishes that $$\left ( \frac{1}{2}-o(1) \right ) n\log n \le τ_3(n) = O\big (n^{\sqrt{2}}\log n \big ),$$ which gives the first non-trivial improvements over the simple lower and upper bounds. Our proof combines various applications of the probabilistic method and a generalisation of the edge-isoperimetric inequality for the hypercube. |
| title | On almost Gallai colourings in complete graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.17334 |