We show that there exists a linear-time algorithm that computes the strong chromatic index of Halin graphs.

Given a finite simple graphG = (V,E) with chromatic number c and chromatic polynomial C(x). Every vertex graph coloring f of G defines an index if (x) satisfying the Poincaré-Hopf theorem [17] ∑ x if (x) = χ(G). As a variant to the index expectation result [19] we prove that E[if (x)] is equal to curvature K(x) satisfying Gauss-Bonnet ∑ xK(x) = χ(G) [16], where the expectation is the average ov...

A new framework is presented for saliency based analysis of naturally complex scenes. In this framework, the fluctuation of local scale shift and the diversity of subtle chromatic scattering are extracted as the environmental saliency: viewer specific visualization of sign patterns to be associated with landmark objects. Being guided by the probability distribution of the scale shift, the scene...

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). The List-Colouring Conjecture (LCC) is that every multigraph G has edgechoosability (list chromatic index) ch’(G) equal to its chromatic index x’(G). In this paper the LCC is proved first for multicircuits, and then, building on results of Peterson and Woodall, for any multigraph G in which ...

In this paper we show upper bounds for the sum and the product of the lower domination parameters and the chromatic index of a graph. We also present some families of graphs for which these upper bounds are achieved. Next, we give a lower bound for the sum of the upper domination parameters and the chromatic index. This lower bound is a function of the number of vertices of a graph and a new gr...

The b-chromatic index φ′(G) of a graph G is the largest integer k such that G admits a proper k-edge coloring in which every color class contains at least one edge incident to edges in every other color class. We give in this work bounds for the b-chromatic index of the direct product of graphs and provide general results for many direct products of regular graphs. In addition, we introduce a l...

The strong chromatic index of a graph G is the minimum number of induced matchings which partition E(G). In 1985, Erd} os and Ne set ril conjectured that the strong chromatic index of every graph of maximum degree is at most 5 4 2. In this paper, we use probabilistic method to prove an asymptotically better result for graphs which do not have a cycle of length 4. Also, we show that our bound is...

A twin edge k-coloring of a graph G is a proper edge coloring of G with the elements of Zk so that the induced vertex coloring in which the color of a vertex v in G is the sum (in Zk) of the colors of the edges incident with v is a proper vertex coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Among the results presented are formulas for th...

An investigation of the chromatic properties of waveguide lenses is described. In general, the focal length of mode-index and Fresnel zone (diffractive) lenses will be a function of wavelength. As a result, these lenses will have high optical quality over only a relatively small wavelength range. A method is given to correct for this chromatic dispersion by forming a hybrid mode-index/diffracti...

A simple derivation of analytic expressions for the chromatic paraxial aberration coefficients of radial gradientindex lenses is presented. By decomposing the transverse chromatic aberration vector of an arbitrary paraxial ray in contributions from refraction at the surfaces and from transfer through the inhomogeneous media of the system, remarkably short formulas for the contributions of trans...