Extending regular edge-colorings of complete hypergraphs

A coloring (partition) of the collection $(\genfrac{}{}{0.0pt}{}{X}{h})$ of all $h$-subsets of a set $X$ is $r$-regular if the number of times each element of $X$ appears in each color class (all sets of the same color) is the same number $r$. We are interested in finding the conditions under which a given $r$-regular coloring of $(\genfrac{}{}{0.0pt}{}{X}{h})$ is extendible to an $s$-regular coloring of $(\genfrac{}{}{0.0pt}{}{Y}{h})$ for $s⩾r$ and $Y⊋X$. The case $\text{◂,▸}h=2,r=s=1$ was solved by Cruse, and due to its connection to completing partial symmetric latin squares, many related problems are extensively studied in the literature, but very little is known for $h⩾3$. The case $r=s=1$ was solved by Häggkvist and Hellgren, settling a conjecture of Brouwer and Baranyai. The cases $h=2$ and $h=3$ were solved by Rodger and Wantland, and Bahmanian and Newman, respectively. In this paper, we completely settle the cases $h=4,\text{◂⩾▸}|Y|⩾4|X|$ and $h=5,\text{◂⩾▸}|Y|⩾5|X|$.     

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.     

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.     

The inverse conductivity problem via the calculus of functions of bounded variation

In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of $BV$ functions. The space of the functions of bounded variation is recommended here as the most appropriate functional space hosting the conductivity profile under reconstruction. For the numerical investigation of the inversion of the inclusion problem, we propose and implement a suitable minimization scheme of an enriched—constructed herein—functional, by exploiting the inner structure of $BV$ space. Finally, we validate and illustrate our theoretical results with numerical experiments.     

The optimal general upper bound for the distinguishing index of infinite graphs

The distinguishing index $D\prime \left(G\right)$ of a graph $G$ is the least cardinal number $d$ such that $G$ has an edge-coloring with $d$ colors, which is preserved only by the trivial automorphism. We prove a general upper bound $D\prime \text{◂≤▸}\left(G\right)\le \mathrm{\Delta }-1$ for any connected infinite graph $G$ with finite maximum degree $\mathrm{\Delta }\ge 3$. This is in contrast with finite graphs since for every $\mathrm{\Delta }\ge 3$ there exist infinitely many connected, finite graphs $G$ with $\text{◂,▸}D\prime \left(G\right)=\mathrm{\Delta }$. We also give examples showing that this bound is sharp for any maximum degree $\mathrm{\Delta }$.     

Edge proximity and matching extension in projective planar graphs

A graph $G$ with at least $2m+2$ vertices is said to be distance $d$ $m$-extendable if, for any matching $M$ of $G$ with $m$ edges in which the edges lie at distance at least $d$ pairwise, there exists a perfect matching of $G$ containing $M$. In this paper we prove that every 5-connected triangulation on the projective plane of even order is distance 3 7-extendable and distance 4 $m$-extendable for any $m$.