CID 2003 Abstracts - Colourings, Independence and Domination

48 abstracts
RAMSEY AND RAINBOW COLOURINGS
Ingo Schiermeyer
Freiberg University of Mining and Technology, Freiberg, Germany
In this talk we consider edge colourings of graphs. For given graphs F 1 , F 2 ,
. . . , F k , k ≥ 2, the Ramsey number r(F 1 , . . . , F k ) is the smallest integer n such
that if we arbitrarily colour the edges of the complete graph of order n with k
colours, then there is always a monochromatic copy of some F i for 1 ≤ i ≤ k.
We will list the Ramsey numbers if the graphs F i are complete or cycles and
report about recent progress on some conjectures of Erdős ([2], [3]).
For given graphs G, H the rainbow number rb(G, H) is the smallest number
m of colours such that if we colour the edges of G with at least m different
colours, then there is always a totally multicoloured or rainbow copy of H.
For various graph classes of H we will list the known rainbow numbers if G
is the complete graph [1] and report about recent progress on the conjecture
of Erdős, Simonovits and Sós on the rainbow numbers rb(K n , C k ) for cycles.
Finally, new results on the rainbow numbers rb(Q n , Q 2 ) for the hypercube Q n
will be presented.
Keywords: edge colouring, Ramsey, rainbow, extremal graphs.
AMS Subject Classification: 05C15, 05C35.
References
[1] I. Schiermeyer, Rainbow colourings, Notices of the South African Mathematical
Society 34 (1) (April 2003) 51–59.
[2] R. Faudree, A. Schelten, I. Schiermeyer, The Ramsey number r(C 7 , C 7 , C 7 ),
Discussiones Mathematicae Graph Theory 23 (2003) 141–158.
[3] I. Schiermeyer, All cycle-complete graph Ramsey numbers r(C m , K 6 ), J.
Graph Theory, to appear.