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.     

Edge Kempe equivalence of regular graph covers

Let $G$ be a finite $d$-regular graph with a proper edge coloring. An edge Kempe switch is a new proper edge coloring of $G$ obtained by switching the two colors along some bichromatic cycle. We prove that any other edge coloring can be obtained by performing finitely many edge Kempe switches, provided that $G$ is replaced with a suitable finite covering graph. The required covering degree is bounded above by a constant depending only on $d$.     

Reconfiguration of connected graph partitions

Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph $G$ and an integer $k\ge 1$, a $k$-district map of $G$ is a partition of $V\left(G\right)$ into $k$ nonempty subsets, called districts, each of which induces a connected subgraph of $G$. A switch is an operation that modifies a $k$-district map by reassigning a subset of vertices from one district to an adjacent district; a 1-switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all $k$-district maps of a graph $G$ under 1-switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is PSPACE-complete to decide whether there exists a sequence of 1-switches that takes a given $k$-district map into another; and NP-hard to find the shortest such sequence (even if a sequence of polynomial lengths is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that take a given $k$-district map into another when the space is connected, and show that these algorithms perform a worst-case optimal number of switches up to constant factors.     

Hamiltonian cycles in k-partite graphs

Chen et al determined the minimum degree threshold for which a balanced $k$-partite graph has a Hamiltonian cycle. We give an asymptotically tight minimum degree condition for Hamiltonian cycles in arbitrary $k$-partite graphs in that all parts have at most $n/2$ vertices (a necessary condition). To do this, we first prove a general result that both simplifies the process of checking whether a graph $G$ is a robust expander and gives useful structural information in the case when $G$ is not a robust expander. Then we use this result to prove that any $k$-partite graph satisfying the minimum degree condition is either a robust expander or else contains a Hamiltonian cycle directly.     

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.     

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.     

Between the enhanced power graph and the commuting graph

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting graph of $G$. Two elements of $G$ are joined in the deep commuting graph if and only if their inverse images in every central extension of $G$ commute. We give conditions for the graph to be equal to either of the enhanced power graph and the commuting graph, and show that automorphisms of $G$ act as automorphisms of the deep commuting graph.