We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this paper we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG repres...

Let 1R be the preorder of embeddability between countable linear orders colored with elements of Rado’s partial order (a standard example of a wqo which is not a bqo). We show that 1R has fairly high complexity with respect to Borel reducibility (e.g. if P is a Borel preorder then P ≤B 1R), although its exact classification remains open.

The visualization has become essential in many application areas. The finite graphs and automata undoubtedly belong to such areas. A problem of a visual image of a directed finite graph has appeared in the study of the road coloring conjecture. Given a finite directed graph, a coloring of its edges turns the graph into finitestate automaton. The visual perception of the structure properties of ...

A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph Pk denotes a path on k vertices. The `-Coloring problem is the problem to decide whether a graph can be colored with at most ` colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8free graphs. This improves a result of Le, Randerath, and Schierme...

Linear scan register allocation is an efficient alternative to the widely used graph coloring approach. We show how this algorithm can be applied to register-constrained architectures like the Intel x86. Our allocator relies on static single assignment form, which simplifies data flow analysis and tends to produce short live intervals. It makes use of lifetime holes and instruction weights to i...

4-Methylimidazole (4-MEI) is a simple nitrogen-containing heterocyclic compound, and recently classified as group 2B by International Agency for Research on Cancer (IARC). Caramel coloring is widely applied food coloring in many food products including coffee, carbonated beverages, beer and wine. In present study, ɣ-irradiation was applied to Caramel Color III to reduce the level of 4-MEI witho...

If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is ∆ if ∆ ≥ 8, the list edge ...

The problem of coloring a set of n intervals (from the real line) with a set of k colors is studied. In such a coloring, two intersecting intervals must receive distinct colors. Our main result is an O(k + n) algorithm for k-coloring a maximum cardinality subset of the intervals, assuming that the endpoints of the intervals are presorted. Previous methods are linear only in n, and assume that k...

A graph coloring algorithm that immediatly colors the vertices taken from a list without looking ahead or changing colors already assigned is called an one-line coloring algorithmm. We study in this paper on-line oriented coloring algorithms. Firstly, we investigate the First-Fit algorithm and we try to characterize the F F-optimall graphs that is the graphs for which there exists a linear orde...