CID 2003 Abstracts - Colourings, Independence and Domination

34 abstracts
ESTIMATION OF CUT-VERTICES IN EDGE-COLOURED
COMPLETE GRAPHS
Adam Idzik
Świȩtokrzyska Academy, Kielce and Polish Academy of Sciences, Warsaw, Poland
Given a k-edge-coloured graph G = (V, E 1 , ..., E k ), we define F i = E \ E i ,
G i = (V, E i ), Ḡi = (V, F i ), where E = ⋃ i∈{1,...,k} Ei and i ∈ {1, ..., k}. Here G i
is a monochromatic subgraph of G and Ḡi is its complement in G.
The following theorem [1] is under discussion.
Let (E 1 , ..., E k ) be a k-edge-colouring of K m (k ≥ 2, m ≥ 4), such that all the
graphs Ḡ1 , · · · , Ḡk are connected.
(i) If one of the subgraphs G 1 , · · · , G k is 2-connected, say G i , then c(Ḡi ) ≤
m − 2 and c(Ḡj ) = 0 for j ≠ i (i, j ∈ {1, ..., k}).
(ii) If none of the graphs G 1 , · · · , G k is 2-connected, and one of them is connected,
say G i , then c(Ḡi ) ≤ 2 (i ∈ {1, ..., k}).
(iii) If none of the graphs G 1 , ..., G k is 2-connected, and one of them is disconnected,
say G i , then c(Ḡi ) ≤ 1 (i ∈ {1, ..., k}).
Keywords: complete graph, connected graph, cut-vertex, edge-colouring.
AMS Subject Classification: 05C35, 05C40, 68R10.
References
[1] A. Idzik, Zs. Tuza, X. Zhu, Cut-vertices in edge-coloured complete graphs,
preprint.