A homomorphism from an oriented graph G to an oriented graph H is a mapping φ from the set of vertices of G to the set of vertices of H such that −−−−−→ φ(u)φ(v) is an arc in H whenever −→uv is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (Reca...

A graph is chromatic-index-critical if it cannot be edge-coloured with ∆ colours (with ∆ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order 18 [18, 31]. In this paper we show that there are no chromatic-inde...

The effective refractive index as a function of vacuum wavelength is approximated by Lagrange interpolation polynomials. The rms value of the chromatic dispersion is then calculated analytically. Itis demonstrated that use of fourth-degree polynomials is far more efficient than the use of second-degree polynomials. The rms value of the chromatic dispersion over the wavelength range (1.25 µm, 1....

The strong chromatic index of a graph G, denoted by χs(G), is the minimum number of colors needed to color its edges so that each color class is an induced matching. In this paper we analyze the asymptotic behavior of this parameter in a random graph G(n, p), for two regions of the edge probability p = p(n). For the dense case, where p is a constant, 0 < p < 1, we prove that with high probabili...

We prove that the game chromatic index of n-wheels is n for n ≥ 6. © 2010 Elsevier B.V. All rights reserved.

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 5 k 5 1 5 m, the subset graph S,(k, 1) is a bipartite graph whose vertices are the kand 1-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that sq(Sm(k, 1 ) ) = (&) a...

If G = (V ′, E) is a graph and H = (V,H) is a graph whose edges can be decomposed into isomorphic copies of G, then we define a k-block colouring of a G−decomposition of H to be an assignment of k colours to the copies of G so that no two copies of G having a vertex in common have the same colour. A G−decomposition of H has chromatic index χ′ = k if it is k block colourable and not k − 1 block ...

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving ®rst, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when ...

In a recent article [5], the authors claim that the distance between the b-chromatic index of a tree and a known upper bound is at most 1. At the same time, in [7] the authors claim to be able to construct a tree where this difference is bigger than 1. However, the given example was disconnected, i.e., actually consisted of a forest. Here, we slightly modify their construction in order to produ...

A multigraph M with maximum degree (M) is called critical, if the chromatic index 0 (M) > (M) and 0 (M ? e) = 0 (M) ? 1 for each edge e of M. The weak critical graph conjecture 1, 7] claims that there exists a constant c > 0 such that every critical multigraph M with at most c (M) vertices has odd order. We disprove this conjecture by constructing critical multigraphs of order 20 with maximum d...