Shannon–Vizing–type problems concerning the upper bound for a distance chromatic index of multigraphs G in terms of the maximum degree ∆(G) are studied. Conjectures generalizing those related to the strong chromatic index are presented. The chromatic d–index and chromatic d–number of paths, cycles, trees and some hypercubes are determined. Among hypercubes, however, the exact order of their gro...

waveguides with low confinement loss, low chromatic dispersion, and low nonlinear effects are used in optical communication systems. optical fibers can also be employed in such systems. besides optical fibers, photonic crystal fibers are also highly suitable transmission media for optical communication systems. in this paper, we introduce two new designs of index-guiding photonic crystal fiber ...

The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for the strong chromatic index of three different products of graphs in terms of the strong chromatic index of each factor. For the Cartesian product of paths, cycles or complete graphs, we derive sharper ...

We prove that the acyclic chromatic number of a graph with maximum degree ∆ is less than 2.835∆4/3+∆. This improves the previous upper bound, which was 50∆4/3. To do so, we draw inspiration from works by Alon, McDiarmid and Reed and by Esperet and Parreau.

Signed graphs are studied since the middle of the last century. Recently, the notion of homomorphism of signed graphs has been introduced since this notion captures a number of well known conjectures which can be reformulated using the definitions of signed homomorphism. In this paper, we introduce and study the properties of some target graphs for signed homomorphism. Using these properties, w...

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding down-chromatic number and derive an upper bound as a function of D(~ G), the maximum number of descendants of a gi...

Greene and Zaslavsky proved that the number of acyclic orientations of a graph G with a unique sink at a given vertex is, up to sign, the linear coefficient of the chromatic polynomial. We give three proofs of this result using pure induction, noncommutative symmetric functions, and an algorithmic bijection. We also prove their result that if e=u0v0 is an edge of G then the number of acyclic or...

Let D be a digraph. The chromatic number χ(D) of D is the smallest number of colors needed to color the vertices of D such that every color class induces an acyclic subdigraph. The girth of D is the length of a shortest directed cycle, or ∞ if D is acyclic. Let G(k, n) be the maximum possible girth of a digraph on n vertices with χ(D) > k. It is shown that G(k, n) ≥ n1/k and G(k, n) ≤ (3 log2 n...

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding down-chromatic number and derive an upper bound as a function of D(~ G), the maximum number of descendants of a gi...

A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20.