A sharp upper bound for the harmonious total chromatic number of graphs and multigraphs

A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest number of colours for it to exist is called the harmonious total chromatic number of $G$, denoted by $h_t(G)$. Here, we give a general upper bound for $h_t(G)$ in terms of the order $n$ of $G$. Our two main results are obvious consequences of the computation of the harmonious total chromatic number of the complete graph $K_n$ and of the complete multigraph $\lambda K_n$, where $\lambda$ is the number of edges joining each pair of vertices of $K_n$. In particular, Araujo-Pardo et al. have recently shown that $\frac{3}{2}n\leq h_t(K_n) \leq \frac{5}{3}n +\theta(1)$. In this paper, we prove that $h_t(K_{n})=\left\lceil \frac{3}{2}n \right\rceil$ except for $h_t(K_{1})=1$ and $h_t(K_{4})=7$; therefore, $h_t(G) \le \left\lceil \frac{3}{2}n \right\rceil$, for every graph $G$ on $n>4$ vertices. Finally, we extend such a result to the harmonious total chromatic number of the complete multigraph $\lambda K_n$ and as a consequence show that $h_t(\mathcal{G})\leq (\lambda-1)(2\left\lceil\frac{n}{2}\right\rceil-1)+\left\lceil\frac{3n}{2}\right\rceil$ for $n>4$, where $\mathcal{G}$ is a multigraph such that $\lambda$ is the maximum number of edges between any two vertices.Comment: 11 pages, 5 figure

Ground state of excitons and charged excitons in a quantum well

A variational calculation of the ground state of a neutral exciton and of positively and negatively charged excitons (trions) in single quantum well is presented. We study the dependance of the correlation energy and of the binding energy on the well width and on the hole mass. Our results are are compared with previous theoretical results and with avalaible experimental data.Comment: 8 pages, 5 figures presented to OECS

Well-width dependence of the ground level emission of GaN/AlGaN quantum wells

We have performed a systematic investigation of GaN/AlGaN quantum wells grown on different buffer layers (either GaN or AlGaN) in order to clarify the role of strain, structural parameters, and built-in field in determining the well-width dependence of the ground level emission energy. We find that identical quantum wells grown on different buffer layers exhibit strong variation of the ground level energy but similar well-width dependence. The data are quantitatively explained by an analytic model based on the envelope function formalism which accounts for screening and built-in field, and by a full self-consistent tight binding model

Spontaneous polarization and piezoelectric field in G a N / A l 0.15 Ga 0.85 N quantum wells: Impact on the optical spectra

We have investigated the effects of the built-in electric field in ${\mathrm{G}\mathrm{a}\mathrm{N}/\mathrm{A}\mathrm{l}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ quantum wells by photoluminescence spectroscopy. The fundamental electron heavy-hole transition redshifts well below the GaN bulk gap for well widths larger than 3 nm for the specific quantum wells investigated and exhibits a concomitant reduction of the intensity with increasing well thickness. The experimental data are quantitatively explained by means of a self-consistent tight-binding model that includes screening (either dielectric or by free-carriers), piezoelectric field and spontaneous polarization field. The impact of the built-in field on the exciton stability is discussed in detail. We demonstrate that the exciton binding energy is substantially reduced by the built-in field, well below the values expected from the quantum size effect in the flat band condition

Graph products and new solutions to Oberwolfach problems

RP5