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# By a k-edge- colouring of a graph we mean any finite partition of the set of its edges into k subsets

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Problems Column 217

Discussiones Mathematicae Graph Theory 25 (2005 ) 217–218

ESTIMATION OF CUT-VERTICES IN EDGE-COLOURED COMPLETE GRAPHS

15 ´Swi¸etokrzyska street, 25–406 Kielce, Poland and

Institute of Computer Science Polish Academy of Sciences 21 Ordona street, 01–237 Warsaw, Poland

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 GV \{v} = (V \ {v}, E ∩ 2V \{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-colouring (E1, · · · , Ek) (Ei ∩ Ej = ∅ for i 6= j; i, j ∈ {1, · · · , k} and Si∈{1,···,k}Ei = E) is denoted by (V, E1, · · · , Ek). The graphs (V, Ei) are called monochromatic subgraphs of (V, E1, · · · , Ek), i ∈ {1, · · · , k}. As usual, by Km we denote the complete graph with m vertices.

Let c(Gi) denote the number of cut-vertices of Gi in a monochromatic subgraph Gi = (V, Ei) of a k-edge-coloured complete graph Km = (V, E1,

· · · , Ek) (i ∈ {1, · · · , k}).

Given a k-edge-coloured graph G = (V, E1, · · · , Ek), we define Fi = E \ Ei, Gi = (V, Ei), ¯Gi = (V, Fi), where E =Si∈{1,···,k}Ei and i ∈ {1, · · · , k}.

Here Gi is a monochromatic subgraph of G and ¯Gi its complement in G.

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218 A. Idzik

Theorem (Idzik, Tuza, Zhu). Let (E1, · · · , Ek) be a k-edge-colouring of Km (k ≥ 2, m ≥ 4), such that all the graphs ¯G1, · · · , ¯Gk are connected.

(i) If one of the subgraphs G1, · · · , Gk is 2-connected, say Gi, then c( ¯Gi) ≤ m − 2 and c( ¯Gj) = 0 for j 6= i (i, j ∈ {1, . . . , k}).

(ii) If none of the graphs G1, · · · , Gk is 2-connected, and one of them is connected, say Gi, then c( ¯Gi) ≤ 2 (i ∈ {1, · · · , k}).

(iii) If none of the graphs G1, · · · , Gk is 2-connected, and one of them is disconnected, say Gi, then c( ¯Gi) ≤ 1 (i ∈ {1, · · · , k}).

Problem. Let (E1, · · · , Ek) be a k-edge-colouring of Km (k ≥ 2, m ≥ 4).

What is the cardinality of the set of the sum of cut-vertices of ¯Gi in the case none of Gi is 2-connected and (a) two of Gi are connected or (b) two of Gi are disconnected and c( ¯Gi) = 1 (i ∈ {1, · · · , k}) ?

Observe that in both cases (a) and (b) all the graphs ¯G1, · · · , ¯Gk are con- nected.

This problem is related to some theorems presented in [1] and [2].

References

[1] J. Bos´ak, A. Rosa and ˘S. Zn´am, On decompositions of complete graphs into factors with given diameters, in: P. Erd˝os and G. Katona, eds., Theory of Graphs, Proceedings of the Colloquium Held at Tihany, Hungary (Academic Press, New York, 1968) 37–56.

[2] A. Idzik and Z. Tuza, Heredity properties of connectedness in edge-coloured complete graphs, Discrete Math. 235 (2001) 301–306.

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