Note on the number of edges in families with linear union-complexity

We give a simple argument showing that the number of edges in the intersection graph $G$ of a family of $n$ sets in the plane with a linear union-complexity is $O(\omega(G)n)$. In particular, we prove $\chi(G)\leq \text{col}(G)< 19\omega(G)$ for intersection graph $G$ of a family of pseudo-discs, which improves a previous bound.Comment: background and related work is now more complete; presentation improve

On the imaginary parts of chromatic root

While much attention has been directed to the maximum modulus and maximum real part of chromatic roots of graphs of order $n$ (that is, with $n$ vertices), relatively little is known about the maximum imaginary part of such graphs. We prove that the maximum imaginary part can grow linearly in the order of the graph. We also show that for any fixed $p \in (0,1)$, almost every random graph $G$ in the Erd\"os-R\'enyi model has a non-real root.Comment: 4 figure

Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.Comment: One result proved in this paper is also present in arXiv:1212.675