We introduce the concept of low rank-width colorings, generalizing the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [25]. We say that a class C of graphs admits low rank-width colorings if there exist functions N : N→ N and Q : N→ N such that for all p ∈ N, every graph G ∈ C can be vertex colored with at most N(p) colors such that the union of any i ≤ p colo...

Let S be a point set in the plane in general position, such that its elements are partitioned into k classes or colors. In this paper we study several variants on problems related to the Erdős-Szekeres theorem about subsets of S in convex position, when additional chromatic constraints are considered. Most of this work was made possible by the Picasso French-Spanish collaboration program and th...

In 1987 Paul Erdős asked me if the Cayley graph defined on Z by a lacunary sequence has necessarily a finite chromatic number. Below is my answer, delivered to him on the spot but neverpublished, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems. 1 The Cayley graph ZΛ 1.1 Let Λ ⊂ N. By definition, the Cayley graph ZΛ is the ...

License plate detection from images is an important step in vehicle license plate recognition for automated transport system. Dark images and complex background make license plate detection problems challenging. In this paper, a novel method of based on rectangle window was proposed to resolve license plate detection problems. The proposed method firstly enhances images using traditional method...

We all know about The Book in which, according to Erdős, God keeps the most elegant proof of each mathematical result. It is interesting to speculate whether every theorem has a “Book proof” (say what about the Four Color Theorem or the Collatz 3n+1 Conjecture?) or whether a “Book proof” of a theorem is unique. I believe many mathematicians have their own private collections of proofs from The ...

The classical result in the theory of random graphs, proved by Erdős and Rényi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of r colors. The goal can be either to create a giant component in e...

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‎let $f$ be a proper $k$-coloring of a connected graph $g$ and‎ ‎$pi=(v_1,v_2,ldots,v_k)$ be an ordered partition of $v(g)$ into‎ ‎the resulting color classes‎. ‎for a vertex $v$ of $g$‎, ‎the color‎ ‎code of $v$ with respect to $pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_pi}(v)=(d(v,v_1),d(v,v_2),ldots,d(v,v_k))$‎, ‎where $d(v,v_i)=min{d(v,x):~xin v_i}‎, ‎1leq ileq k$‎. ‎if‎ ‎distinct...

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assump...

Colorizing two–dimensional (2-D) barcodes is a promising modality for improvement on design and data capacity. The number of colors employed in recent 2-D color barcodes is limited less than 8. To increase the number of recognizable colors, we propose a color recognition method which classifies 64 colors, using an extended color space and pattern recognition techniques. The proposed method empl...