We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least (bn/3c 4 ) + (b(n+1)/3c 4 ) + (b(n+2)/3c 4 ) . Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebras framework together with some additional stability arguments. This problem is equivalent to some s...

In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: k-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian [18]. In this coloring, the set S(v) of colors used by edges incident to a vertex v does not intersect S(u) on more than k colors when u and v are adjacent. We provide some sharp upper and lower bounds for χk-int for several ...

The visual systems in primates are organized around orientation with a rich set of long-range horizontal connections. We abstract this from a differential-geometric perspective, and introduce the covariant derivative of frame fields as a general framework for early vision. This paper overviews our research showing how curve detection, texture, shading, color (hue), and stereo can be unified wit...

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring usin...

This paper investigates into the added value of color information for edge extraction and straight edge segment matching between stereo views. For edge extraction in color images different methods proposed in the literature are evaluated and compared, paying special attention to significance and completeness of the obtained edge-map. To find related edge segment pairs in different views, we app...

Abstract: This paper presents a new general type-2 fuzzy logic method for edge detection applied to color format images. The proposed algorithm combines the methodology based on the image gradients and general type-2 fuzzy logic theory to provide a powerful edge detection method. General type-2 fuzzy inference systems are approximated using the α-planes approach. The edge detection method is te...

Let G be an edge-colored copy of Kn, where each color appears on at most n/2 edges (the edgecoloring is not necessarily proper). A rainbow spanning tree is a spanning tree of G where each edge has a different color. Brualdi and Hollingsworth [4] conjectured that every properly edge-colored Kn (n ≥ 6 and even) using exactly n−1 colors has n/2 edge-disjoint rainbow spanning trees, and they proved...

In the edge precoloring extension problem we are given a graph with some of the edges having a preassigned color and it has to be decided whether this coloring can be extended to a proper k-edge-coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We s...

We consider constrained proper edge colorings of the following type: Given a positive integer j and a family F of connected graphs on 3 or more vertices, we require that the subgraph formed by the union of any j color classes has no copy of any member of F . This generalizes some well-known types of colorings such as acyclic edge colorings, distance-2 edge colorings, low treewidth edge coloring...