Pim van 't Hof
University of Bergen, Norway
Ramsey numbers for line graphs and perfect graphs
For any graph class ${\cal G}$ and any two positive integers $i$ and
$j$, the Ramsey number $R_{\cal G}(i,j)$ is the smallest integer $p$
such that every graph in ${\cal G}$ on $p$ vertices has a clique of
size $i$ or an independent set of size $j$. Ramsey numbers for general
graphs are notoriously hard to determine, and exact values have been
established only for very small integers $i$ and $j$. An exact formula
is known for determining all Ramsey numbers for planar graphs, whereas
for claw-free graphs there are infinitely many Ramsey numbers that are
as difficult to determine as for arbitrary graphs. No other graph
classes seem to have been studied in this context.
We give exact formulas for determining all Ramsey numbers for various
classes of graphs. Our main result is an exact formula for all Ramsey
numbers for line graphs, which form a large subclass of claw-free
graphs. We obtain this by proving a tight upper bound on the number of
edges in an arbitrary graph in terms of its maximum degree and maximum
matching size. As complementary results, we determine all Ramsey
numbers for perfect graphs and for several subclasses of perfect
graphs.
This is joint work with R\'{e}my Belmonte, Pinar Heggernes and Reza
Saei.