Colour the edges of a \(K_{24n+1}\) by \(12\) colours so that every vertex in every colour has degree \(2n\). Is there a totally multicoloured \(C_4\) (i.e. every edge gets a different colour)? Here we answer in the affirmative to this question. In [1] P. Erdős stated the same problem for \(K_{12n+1}\) and \(6\) colours, it was settled in [2].
In this paper we follow the terminology and symbols of [3]. We assume the complete graph \(K_{24n+1}\) to have the vertex-set \(V=V(K_{24n+1}) = \{1, 2, \ldots, 24n+1\}\).
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