Mario Lomelí

Universidad Autónoma de San Luis Potosí | UASLP · Instituto de Física

Let P be a finite set of points in general position in the plane. The disjointness graph of segments D(P) of P is the graph whose vertices are all the closed straight line segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. As usual, we use \(\chi (D(P))\) to denote the chromatic number of D(P), and use...

Let $P$ be a finite set of points in general position in the plane. The disjointness graph of segments $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in $D(P)$ if and only if they are disjoint. As usual, we use $\chi(D(P))$ to denote the chromatic number of $D(P)$...

The maximum rectilinear crossing number of a graph $G$ is the maximum number of crossings in a good straight-line drawing of $G$ in the plane. In a good drawing any two edges intersect in at most one point (counting endpoints), no three edges have an interior point in common, and edges do not contain vertices in their interior. A spider is a subdiv...

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different colors. In this paper we show that if $P$ is the point configuration known as the double chain, with $k$ poin...

This paper has been withdrawn by the authors due to a crucial error in the proof of the main result.

Let $P$ be a set of $n$ points on the plane in general position. We say that a set $\Gamma$ of convex polygons with vertices in $P$ is a convex decomposition of $P$ if: Union of all elements in $\Gamma$ is the convex hull of $P,$ every element in $\Gamma$ is empty, and for any two different elements of $\Gamma$ their interiors are disjoint. A minim...

It is shown that if a simple Euclidean arrangement of n pseudolines has no (≥ 5)-gons, then it has exactly n − 2 triangles and (n − 2)(n − 3)/2 quadrilaterals. We also describe how to construct all such arrangements, and as a consequence we show that they are all stretchable.

We find a lower bound for the proportion of face boundaries of an embedded graph that are nearly–light (that is, they have bounded length and at most one vertex of large degree). As an application, we show that every sufficiently large k–crossing–critical graph has crossing number at most 2k + 23.