  The main goal of this paper is to formalize and explore a connection between
chromatic properties of graphs with geometric representations and competitive
analysis of on-line algorithms, which became apparent after the recent
construction of triangle-free geometric intersection graphs with arbitrarily
large chromatic number due to Pawlik et al. We show that on-line graph coloring
problems give rise to classes of game graphs with a natural geometric
interpretation. We use this concept to estimate the chromatic number of graphs
with geometric representations by finding, for appropriate simpler graphs,
on-line coloring algorithms using few colors or proving that no such algorithms
exist.
  We derive upper and lower bounds on the maximum chromatic number that
rectangle overlap graphs, subtree overlap graphs, and interval filament graphs
(all of which generalize interval overlap graphs) can have when their clique
number is bounded. The bounds are absolute for interval filament graphs and
asymptotic of the form $(\log\log n)^{f(\omega)}$ for rectangle and subtree
overlap graphs, where $f(\omega)$ is a polynomial function of the clique number
and $n$ is the number of vertices. In particular, we provide the first
construction of geometric intersection graphs with bounded clique number and
with chromatic number asymptotically greater than $\log\log n$.
  We also introduce a concept of $K_k$-free colorings and show that for some
geometric representations, $K_3$-free chromatic number can be bounded in terms
of clique number although the ordinary ($K_2$-free) chromatic number cannot.
Such a result for segment intersection graphs would imply a well-known
conjecture that $k$-quasi-planar geometric graphs have linearly many edges.
