For smooth convex disks $A$, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes $G^{text{hom}}(A)$ and $G^{text{sim}}(A)$ of intersection graphs that can be obtained from homothets and similarities of $A$, resp
ectively. Namely, we prove that $G^{text{hom}}(A)=G^{text{hom}}(B)$ if and only if $A$ and $B$ are affine equivalent, and $G^{text{sim}}(A)=G^{text{sim}}(B)$ if and only if $A$ and $B$ are similar.
In 1960, Asplund and Grunbaum proved that every intersection graph of axis-parallel rectangles in the plane admits an $O(omega^2)$-coloring, where $omega$ is the maximum size of a clique. We present the first asymptotic improvement over this six-deca
de-old bound, proving that every such graph is $O(omegalogomega)$-colorable and presenting a polynomial-time algorithm that finds such a coloring. This improvement leads to a polynomial-time $O(loglog n)$-approximation algorithm for the maximum weight independent set problem in axis-parallel rectangles, which improves on the previous approximation ratio of $O(frac{log n}{loglog n})$.
It is proved that triangle-free intersection graphs of $n$ L-shapes in the plane have chromatic number $O(loglog n)$. This improves the previous bound of $O(log n)$ (McGuinness, 1996) and matches the known lower bound construction (Pawlik et al., 2013).
Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of
the poset, admitting constant response time for queries of the form is $x<y$?. This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a very new one. We determine precisely which values of dimension/boolean dimension/local dimension imply that the two other parameters are bounded.
