This paper is an expository piece on edge-chromatic graph theory. The central theorem in this subject is that of Vizing. We shall then explore the properties of graphs where Vizing’s upper bound on the chromatic index is tight, and graphs where the lower bound is tight. Finally, we will look at a few generalizations of Vizing’s Theorem, as well as some related conjectures.

xxx Working notes. Three main things remain to do. (1) Either include Tamar’s argument [now in separate document] for Theorem 5, or replace by some better construction for worst-case upper bound. (2) Put in the average-case lower bound calculus calculations needed in section 6. (3) General version of the worst-case lower bound method in section 7.3. A fourth thing one might contemplate: (4) Is ...

In this paper, stability of slopes with a shallow foundation located on top including the shape of the failure mechanism is investigated in static and seismic states. This is a three dimensional problem which is analyzed two dimensionally because 3D analyses are complex and appropriate softwares are not available. The conventional method of bearing capacity evaluation of foundations on top of ...

Abstract In this article, we provide an explicit upper bound for $h_K \mathcal {R}_K d_K^{-1/2}$ which depends on effective constant in the error term of Ideal Theorem.

Uncertain switched linear systems are known as an important class of control systems. Performance of these systems is affected by uncertainties and its stabilization is a main concern of recent studies. Existing work on stabilization of these systems only provides asymptotical stabilization via designing switching strategy and state-feedback controller. In this paper, a new switching strate...

We prove a sampling discretization theorem for the square norm of functions from finite dimensional subspace satisfying Nikol'skii's inequality with an upper bound on number points order dimension subspace.

In Morton and Silverman [8], the authors give an upper bound on the possible primitive periods of periodic points for rational maps on P (and also automorphisms of P ). I generalize their main theorem to nondegenerate morphisms on smooth, irreducible projective varieties and provide a similar upper bound on primitive periods based on good reduction information for these more general dynamical s...

Deformation of the material during cyclic expansion extrusion (CEE) is investigated using upper-bound theorem. The analytical approximation of forming loads agrees very well with the FEM results for different amounts of chamber diameter, friction factor and also for lower die angles. However, the difference between analytical and numerical solution increases at higher die angles which are expla...

In this article we introduce combinatorial multi-color discrepancies and generalize several classical results from 2–color discrepancy theory to c colors (c ≥ 2). We give a recursive method that constructs c–colorings from approximations of 2–color discrepancies. This method works for a large class of theorems like the ‘six standard deviations’ theorem of Spencer, the Beck–Fiala theorem, the re...

In this section we show how the refined upper bound on the regret of the EXP algorithm proved using the potential function approach (KL divergence) also gives us a better bound for the expert game setup with bandit feedback. Last lecture we showed how in the case of expert prediction with bandit feedback using the Exp3 algorithm, the regret is upper bounded by T 2/3n1/3 using a rough upper boun...