On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We prove upper bounds for the star chromatic index of complete bipartite graphs; in particular we obtain tight upper bounds for the case when one part has size at most $3$. We also consider bipartite graphs $G$ where all vertices in one part have maximum degree $2$ and all vertices in the other part has maximum degree $b$. Let $k$ be an integer ($k\geq 1$), we prove that if $b=2k+1$ then $\chi'_{st}(G) \leq 3k+2$; and if $b=2k$, then $\chi'_{st}(G) \leq 3k$; both upper bounds are sharp. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most $6$; in particular we settle this conjecture for cubic Halin graphs.Comment: 18 page

Spectrum radial velocity analyser (SERVAL). High-precision radial velocities and two alternative spectral indicators

Context: The CARMENES survey is a high-precision radial velocity (RV) programme that aims to detect Earth-like planets orbiting low-mass stars. Aims: We develop least-squares fitting algorithms to derive the RVs and additional spectral diagnostics implemented in the SpEctrum Radial Velocity Analyser (SERVAL), a publicly available python code. Methods: We measured the RVs using high signal-to-noise templates created by coadding all available spectra of each star.We define the chromatic index as the RV gradient as a function of wavelength with the RVs measured in the echelle orders. Additionally, we computed the differential line width by correlating the fit residuals with the second derivative of the template to track variations in the stellar line width. Results: Using HARPS data, our SERVAL code achieves a RV precision at the level of 1m/s. Applying the chromatic index to CARMENES data of the active star YZ CMi, we identify apparent RV variations induced by stellar activity. The differential line width is found to be an alternative indicator to the commonly used full width half maximum. Conclusions: We find that at the red optical wavelengths (700--900 nm) obtained by the visual channel of CARMENES, the chromatic index is an excellent tool to investigate stellar active regions and to identify and perhaps even correct for activity-induced RV variations.Comment: 13 pages, 13 figures. A&A in press. Code is available at https://github.com/mzechmeister/serva