Suppose $n\equiv 1(\mathrm{mod}m)$. We say that an edge-colouring of $K_n$ using $m$ colours is balanced if every vertex sees exactly $\lfloor n/m\rfloor$ many edges of each colours. For which graphs $G$ is it true that, if $m=e(G)$, for all large $n\equiv 1(\mathrm{mod}m)$, every balanced edge-colouring of $K_n$ with $m$ colours contains a rainbow copy of $G$? (That is, a subgraph isomorphic to $G$ where each edge receives a different colour.)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let me denote (\ell:=e(G)) [[nomath]](so $\ell=m$)[[/nomath]]. When (n\equiv 1\pmod \ell), write [ n=1+k\ell\qquad\text{so}\qquad k=\frac{n-1}{\ell}. ] “Balanced” then means: for every vertex $v$ and every colour $c$, (\deg_c(v)=k). Equivalently, each colour class is a spanning $k$-regular graph, and these (\ell) regul…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.