Lower bounds for on-line graph colorings

We propose two strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use $2\log_2 n - 10$ colors, where $n$ is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs graphs that contain neither $C_3$ nor $C_5$ as a subgraph and forces $\Omega(\frac{n}{\log n}^\frac{1}{3})$ colors. The best known on-line coloring algorithm for these graphs uses $O(n^{\frac{1}{2}})$ colors

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

Dynamic Coloring of Unit Interval Graphs with Limited Recourse Budget

In this paper we study the problem of coloring a unit interval graph which changes dynamically. In our model the unit intervals are added or removed one at the time, and have to be colored immediately, so that no two overlapping intervals share the same color. After each update only a limited number of intervals are allowed to be recolored. The limit on the number of recolorings per update is called the recourse budget. In this paper we show, that if the graph remains k-colorable at all times, the updates consist of insertions only, and the final instance consists of n intervals, then we can achieve an amortized recourse budget of $\u1d4aa({k⁷ log n})$ while maintaining a proper coloring with k colors. This is an exponential improvement over the result in [Bartłomiej Bosek et al., 2020] in terms of both k and n. We complement this result by showing the lower bound of $Ω(n)$ on the amortized recourse budget in the fully dynamic setting. Our incremental algorithm can be efficiently implemented. As an additional application of our techniques we include a new combinatorial result on coloring unit circular arc graphs. Let L be the maximum number of arcs intersecting in one point for some set of unit circular arcs $\u1d49c$. We show that if there is a set $\u1d49c'$ of non-intersecting unit arcs of size $L²-1$ such that $\u1d49c ∪ \u1d49c'$ does not contain L+1 arcs intersecting in one point, then it is possible to color $\u1d49c$ with L colors. This complements the work on circular arc coloring [Belkale and Chandran, 2009; Tucker, 1975; Valencia-Pabon, 2003], which specifies sufficient conditions needed to color $\u1d49c$ with L+1 colors or more

Disparity map generation based on trapezoidal camera architecture for multiview video

Visual content acquisition is a strategic functional block of any visual system. Despite its wide possibilities, the arrangement of cameras for the acquisition of good quality visual content for use in multi-view video remains a huge challenge. This paper presents the mathematical description of trapezoidal camera architecture and relationships which facilitate the determination of camera position for visual content acquisition in multi-view video, and depth map generation. The strong point of Trapezoidal Camera Architecture is that it allows for adaptive camera topology by which points within the scene, especially the occluded ones can be optically and geometrically viewed from several different viewpoints either on the edge of the trapezoid or inside it. The concept of maximum independent set, trapezoid characteristics, and the fact that the positions of cameras (with the exception of few) differ in their vertical coordinate description could very well be used to address the issue of occlusion which continues to be a major problem in computer vision with regards to the generation of depth map