Abstract Structural colors have been regarded as an ideal alternative to pigments because of the advantages environmental friendliness, resistance fading, especially in fields textile dyeing and printing, which is highly polluted traditional ways needs be upgraded. They are generated by interaction between micro/nano structure material surface incident light, closely related sizes periods struc...

Properly colored cycles in edge-colored graphs are closely related to directed oriented graphs. As an analogy of the well-known Caccetta–Häggkvist Conjecture, we study existence properly bounded length graph. We first prove that for all integers s and t with ≥ 2 , every graph G no K contains a spanning subgraph H which admits orientation D such cycle is . Using this result, show r 4 if Conjectu...

The study of problems modeled by edge-colored graphs has given rise to important developments during the last few decades. For instance, the investigation of spanning trees for graphs provide important and interesting results both from a mathematical and an algorithmic point of view (see for instance [1]). From the point of view of applicability, problems arising in molecular biology are often ...

Abstract We consider maximum properly edge-colored trees in edge-colored graphs Gc. We also consider the problem where, given a vertex r, determine whether the graph has a spanning tree rooted at r, such that all root-to-leaf paths are properly colored. We consider these problems from graphtheoretic as well as algorithmic viewpoints. We prove their optimization versions to be NP-hard in general...

Let G be a planar graph with n vertices whose vertex set is partitioned into subsets V0, . . . , Vk−1 for some positive integer 1 ≤ k ≤ n and let S be a set of n distinct points in the plane partitioned into subsets S0, . . . , Sk−1 with |Vi| = |Si| (0 ≤ i ≤ k − 1). This paper studies the problem of computing a crossing-free drawing of G such that each vertex of Vi is mapped to a distinct point...