A former conjecture of Burr and Rosta [l], extending a conjecture of Erd6s [2], asserted that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graph G which are monochromatic is at least the proportion found in a random eolouring. It is now known that the conjecture fails for some graphs G, including G = Kp for p>4.We investigate for which graphs G the conjecture holds. Our main result is that the conjecture fails if G contains K 4 as a subgraph, and in particular it fails for almost all graphs.
We determine, to within a constant factor, the maximum size of a digraph that does not contain a topological complete digraph DK p of order p. Let t 1 ( p) be defined for positive p bywhere D denotes a digraph. We show that 1 16 p 2 < t 1 ( p) ≤ 44 p 2 . We also obtain results for containing topological tournaments, and a Turán-type result for containing a topological transitive tournament and a transitive tournament.
We consider the function χ(Gk),
defined to be the smallest number of colours that can
colour a graph G in such a way that no vertices of distance
at most k receive the same
colour. In particular we shall look at how small a value this function
can take in terms of
the order and diameter of G. We get general bounds for this and
tight
bounds for the cases k=2 and k=3.
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