Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all three colours. What happens for more colours: if  $$k$$k-colour the edges of the complete graph, with each colour class connected, how many of the $$\left( {\begin{array}{c}k\\ 3\end{array}}\right) $$k3 triples of colours must appear as triangles? In this note  show that the `obvious' conjecture, namely that there are always at least $$\left( {\begin{array}{c}k-1\\ 2\end{array}}\right) $$k-12 triples, is not correct.  determine the minimum asymptotically. This ansrs a question of Johnson.  also give some results about the analogous problem for hypergraphs, and  make a conjecture that  believe is the `right' generalisation of Gallai's theorem to hypergraphs.
['Imre Leader', 'Ta Sheng Tan']
I am writing a research paper on a idea Connected Colourings of Complete Graphs and Hypergraphs
[{"from": "human", "value": "I am writing a research paper on a idea Connected Colourings of Complete Graphs and Hypergraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Gallai's colouring theorem states that if the edges of a complete graph are 3-coloured, with each colour class forming a connected (spanning) subgraph, then there is a triangle that has all three colours. What happens for more colours: if  $$k$$k-colour the edges of the complete graph, with each colour class connected, how many of the $$\\left( {\\begin{array}{c}k\\\\ 3\\end{array}}\\right) $$k3 triples of colours must appear as triangles? In this note  show that the `obvious' conjecture, namely that there are always at least $$\\left( {\\begin{array}{c}k-1\\\\ 2\\end{array}}\\right) $$k-12 triples, is not correct.  determine the minimum asymptotically. This ansrs a question of Johnson.  also give some results about the analogous problem for hypergraphs, and  make a conjecture that  believe is the `right' generalisation of Gallai's theorem to hypergraphs."}]