Abstract
A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph
$K_n$
arrows a small graph H, then every ‘dense’ subgraph of
$K_n$
also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if
$n = 3t+r$
where
$r \in \{0,1,2\}$
and G is an n-vertex graph with
$\delta(G) \ge (3n-1)/4$
, then for every 2-edge-colouring of G, either there are cycles of every length
$\{3, 4, 5, \dots, 2t+r\}$
of the same colour, or there are cycles of every even length
$\{4, 6, 8, \dots, 2t+2\}$
of the samecolour.
Our result is tight in the sense that no longer cycles (of length
$>2t+r$
) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every
$(3t-1)$
-vertex graph G with minimum degree larger than
$3|V(G)|/4$
arrows the path
$P_{2n}$
with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for
$n=3t+r$
where
$r \in \{0,1,2\}$
and every n-vertex graph G with
$\delta(G) \ge 3n/4$
, in each 2-edge-colouring of G there exists a monochromatic cycle of length at least
$2t+r$
.
Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree