A colored mixed graph has vertices linked by both colored arcs and colored edges. The chromatic number of such a graph G is defined as the smallest order of a colored mixed graph H such that there exists a (arc-color preserving) homomorphism from G to H . We study in this paper the colored mixed chromatic number of planar graphs, partial 2-trees and outerplanar graphs with given girth.

We deal with different algorithmic questions regarding properly arc-colored s-t trails, paths and circuits in arc-colored digraphs. Given an arc-colored digraph D with c ≥ 2 colors, we show that the problem of determining the maximum number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time. Surprisingly, we prove that the determination of a properly arc-colored s-...

Today there are many techniques for web service compositions. Evaluation of quality parameters has great impact on evaluation of final product. BPEL is one of those techniques that several researches have been done on its evaluation. However, there are few researches on evaluation of QoS in eflow. This research tries to evaluate performance and reliability of eflow and BPEL through mapping them...

The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and directed graphs. Gutin et al. [Discrete Applied Math 217 (2016)] generalized these results by proving that CPP on c-edge-colored graphs is polynomial time solvable for every c ≥ 2. In CPP on weighted edge-colored graphs G, we wish to find a minimum weight properly colored closed walk containing all edges...

Sufficient degree conditions for the existence of properly edge-colored cycles and paths in edge-colored graphs, multigraphs and random graphs are inverstigated. In particular, we prove that an edgecolored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to d 2 e has properly edge-colored cycles of all possible lengths, including hamiltonian c...

An edge-colored graph H is properly colored if no two adjacent edges of H have the same color. In 1997, J. Bang-Jensen and G. Gutin conjectured that an edgecolored complete graph G has a properly colored Hamilton path if and only if G has a spanning subgraph consisting of a properly colored path C0 and a (possibly empty) collection of properly colored cycles C1, C2, . . . , Cd such that V (Ci) ...

We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertice coalesce. The rule that governs the change of color involves a parameter x. When x = 1/2, the colored coalescent process ca...

growing demand of invisible braces byesthetically conscious patients has led to remarkable inventions in materials for esthetic labial archwires. archwires with excellent optical clarity and mechanical properties comparable to conventional archwires have been manufactured by almost all the leading companies of orthodontic products in the past two decades, but their clinical use is still limited...

We characterize edge-colored graphs in which every edge belongs to some properly colored cycle. We obtain our result by applying a characterization of 1-extendable graphs.

We show that the 2-edge-colored chromatic number of a class of simple graphs is bounded if and only if the acyclic chromatic number is bounded for this class. Recently, the CSP dichotomy conjecture has been reduced to the case of 2-edge-colored homomorphism and to the case of 2-vertex-colored homomorphism. Both reductions are rather long and follow the reduction to the case of oriented homomorp...