Estimation of cut-vertices in edge-coloured complete graphs

All graphs considered here are finite simple graphs, i.e., graphs without loops, multiple edges or directed edges. For a graph G = (V, E), where V is a vertex set and E is an edge set, we write sometimes V (G) for V and E(G) for E to avoid ambiguity. We shall write G \ v instead of G V \{v} = (V \ {v}, E ∩ 2 V \{v}), the subgraph induced by V \ {v}. A vertex v ∈ V (G) is called a cut-vertex of G if G is connected and G \ v is not. By a k-edge-colouring of a graph we mean any finite partition of the set of its edges into k subsets. A graph (V, E) with a given k-edge-we denote the complete graph with m vertices. Let c(G i) denote the number of cut-vertices of G i in a monochromatic subgraph G i = (V, E i) of a k-edge-coloured complete graph K m