Algebraic techniques for least squares problems in commutative quaternionic theory

Due to the rise of commutative quaternion in Hopfield neural networks, digital signal, and image processing, one encounters the approximate solution problems of the commutative quaternion linear equations $AX\approx B$ and $AXC\approx B$. This paper, by means of real representation and complex representation of commutative quaternion matrices, introduces concepts of norms of commutative quaternion matrices and derives two algebraic techniques for finding solutions of least squares problems in commutative quaternionic theory.     

t-Cores for ( Δ + t )-edge-coloring

We extend the edge-coloring notion of core (subgraph induced by the vertices of maximum degree) to $t$-core (subgraph induced by the vertices $v$ with ), and find a sufficient condition for $\left(\mathrm{\Delta }+t\right)$-edge-coloring. In particular, we show that for any $t\ge 0$, if the $t$-core of $G$ has multiplicity at most $t+1$, with its edges of multiplicity $t+1$ inducing a multiforest, then $\chi \prime \text{◂≤▸}\left(G\right)\le \mathrm{\Delta }+t$. This extends previous work of Ore, Fournier, and Berge and Fournier. A stronger version of our result (which replaces the multiforest condition with a vertex-ordering condition) generalizes a theorem of Hoffman and Rodger about cores of $\mathrm{\Delta }$-edge-colorable simple graphs. In fact, our bounds hold not only for chromatic index, but for the fan number of a graph, a parameter introduced by Scheide and Stiebitz as an upper bound on chromatic index. We are able to give an exact characterization of the graphs $H$ such that $\text{◂...▸}\text{Fan}\left(G\right)\le \text{◂+▸}\mathrm{\Delta }\left(G\right)+t$ whenever $G$ has $H$ as its $t$-core.     

Proper orientation number of triangle-free bridgeless outerplanar graphs

An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation proper if neighboring vertices have different in-degrees. The proper orientation number of a graph $G$, denoted by $\overrightarrow{\chi }\left(G\right)$, is the minimum maximum in-degree of a proper orientation of $G$. Araujo et al asked whether there is a constant $c$ such that $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le c$ for every outerplanar graph $G$ and showed that $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le 7$ for every cactus $G$. We prove that $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le 3$ if $G$ is a triangle-free 2-connected outerplanar graph and $\text{◂≤▸}\overrightarrow{\chi }\left(G\right)\le 4$ if $G$ is a triangle-free bridgeless outerplanar graph.