Colorings versus list colorings of uniform hypergraphs

Let $r$ be an integer with $r\ge 2$ and $G$ be a connected $r$-uniform hypergraph with $m$ edges. By refining the broken cycle theorem for hypergraphs, we show that if $k>\text{◂/▸}\frac{m-1}{\ln (1+\sqrt{2})}\approx \text{◂⋅▸}1.135\left(m-1\right)$, then the $k$-list assignment of $G$ admitting the fewest colorings is the constant list assignment. This extends the previous results of Donner, Thomassen, and the current authors for graphs.     

An Ore-type condition for large k-factor and disjoint perfect matchings

Win conjectured that a graph $G$ on $n$ vertices contains $k$ disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least $n+k-2$, where $n$ is even and $n\ge k+2$. In this paper, we prove that Win's conjecture is true for $k\ge n∕2$, where $n$ is sufficiently large. To show this result, we prove a theorem on $k$-factor in a graph under some Ore-type condition. Our main tools include Tutte's $k$-factor theorem, the Karush-Kuhn-Tucker theorem on convex optimization and the solution to the long-standing 1-factor decomposition conjecture.     

A variant of the Erdős-Sós conjecture

A well-known conjecture of Erdős and Sós states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor 2m/3\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (a) for every $m$ there is a $g\left(m\right)$ such that the weakening of the conjecture obtained by replacing the first $m$ by $g\left(m\right)$ holds, and (b) there is a such that the weakening of the conjecture obtained by replacing $\lfloor 2m/3\rfloor$ by $\text{◂⋅▸}\left(1-\gamma \right)m$ holds.     

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.