Stars and bunches in planar graphs. Part I: Triangulations

Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We prove a theorem on the structure of plane triangulations in terms of stars and bunches. The result states that a plane triangulation contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch. \u

Stars and bunches in planar graphs. Part II: General planar graphs and colourings

Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We first prove a theorem on the structure of plane graphs in terms of stars and bunches. The result states that a plane graph contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch. \u

(k,1)-coloring of sparse graphs

AbstractA graph G is (k,1)-colorable if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with maximum average degree less than 10k+223k+9 admits a (k,1)-coloring, where k≥2. In particular, every planar graph with girth at least 7 is (2,1)-colorable, while every planar graph with girth at least 6 is (5,1)-colorable. On the other hand, when k≥2 we construct non-(k,1)-colorable graphs whose maximum average degree is arbitrarily close to 14k4k+1

Reconfiguration of list edge-colorings in a graph

11th International Symposium, WADS 2009, Banff, Canada, August 21-23, 2009. ProceedingsWe study the problem of reconfiguring one list edge-coloring of a graph into another list edge-coloring by changing one edge color at a time, while at all times maintaining a list edge-coloring, given a list of allowed colors for each edge. First we show that this problem is PSPACE-complete, even for planar graphs of maximum degree 3 and just six colors. Then we consider the problem restricted to trees. We show that any list edge-coloring can be transformed into any other under the sufficient condition that the number of allowed colors for each edge is strictly larger than the degrees of both its endpoints. This sufficient condition is best possible in some sense. Our proof yields a polynomial-time algorithm that finds a transformation between two given list edge-colorings of a tree with n vertices using O(n [superscript 2]) recolor steps. This worst-case bound is tight: we give an infinite family of instances on paths that satisfy our sufficient condition and whose reconfiguration requires Ω(n [superscript 2]) recolor steps

Formation of submicron relief structures on the surface of sapphire substrates

An analysis of technologies that allow creating microrelief structures on the surface of sapphire substrates has been carried out. It is shown that the most effective method of forming relief structures with submicron dimensions is ion beam   etching through a protective mask formed by photolithography. The main problems in creating a microrelief on the surface of sapphire substrates are the removal of static electric charge in the process of ion beam  etching of the substrates, as well as obtaining a protective mask with windows of specified sizes, through which etching of the sapphire substrate is performed

On acyclic colorings of planar graphs

AbstractThe conjecture of B. Grünbaum on existing of admissible vertex coloring of every planar graph with 5 colors, in which every bichromatic subgraph is acyclic, is proved and some corollaries of this result are discussed in the present paper