Multicolour Turán problems
A colouring of a multigraph \(G\) with \(k\) colours is a decomposition of its egde multiset into \(k\) colour classes. A submultigraph \(H\) of \(G\) is called multicoloured or rainbow if its edges all have distinct colours. The parameter \(\text{ex}_k(n,H)\) is defined as the maximum number of edges in a multigraph of order \(n\), that has a \(k\)-colouring not containing a multicoloured copy of \(H\). This parameter provides a generalisation of the Turán number \(\text{ex}(n,H)\) that denotes the maximum number of edges in a graph on \(n\) vertices not containing a copy of \(H\). Some relationships of \(\text{ex}_k(n,H)\) and \(\text{ex}(n,H)\) are presented. The exact values of the investigated parameter are obtained for sufficiently large complete graphs, 3-colour-critical graphs, some bipartite graphs (particularly for \(C_4\)).
scientific article; zbMATH DE number 2107226