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 $\mathrm{\Delta }$.     

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.     

Graph homomorphism reconfiguration and frozen H-colorings

We say that a graph $F$ strongly arrows a pair of graphs $\left(G,H\right)$ and write $\text{◂→▸}F\stackrel{\text{ind}}{\to }\left(G,H\right)$ if any coloring of its edges with red and blue leads to either a red $G$ or a blue $H$ appearing as induced subgraphs of $F$. The induced Ramsey number, $\text{◂...▸}\text{IR}\left(G,H\right)$, is defined as $\text{◂lim▸}\min \text{◂{}▸}\{|V\left(F\right)|:\text{◂→▸}F\stackrel{\text{ind}}{\to }\left(G,H\right)\}$. We consider the connection between the induced Ramsey number for a pair of two connected graphs $\text{◂...▸}\text{IR}\left(G,H\right)$ and the induced Ramsey number for multiple copies of these graphs $\text{IR}\text{◂()▸}\left(sG,tH\right)$, where $xG$ denotes the pairwise vertex-disjoint union of $x$ copies of $G$. It is easy to see that if $\text{◂→▸}F\stackrel{\text{ind}}{\to }\left(G,H\right)$, then $\text{◂⋅▸}\left(s+t-1\right)F\stackrel{\text{ind}}{\to }\text{◂()▸}\left(sG,tH\right)$. This implies that $$\text{◂...▸}\text{IR}\text{◂≤▸}\text{◂()▸}\left(sG,tH\right)\le \left(s+t-1\right)\text{IR}\left(G,H\right).$$ For several specific graphs, such as a path on three vertices vs a complete multipartite graph, a matching vs a complete graph, or a matching vs another matching, it is known that the above inequality is tight. On the other hand, the inequality is also strict for some graphs. However, even in the simplest case when $H$ is an edge and $t=2$, it is not known for what $G$ and for what $s$ the above inequality is tight. We show that it is tight if $G$ is connected and $s$ is at least as large as the order of $G$. In addition, we make further progress in determining induced Ramsey numbers for multiple copies of graphs, such as paths and triangles.     

A characterization of (4,2)-choosable graphs

A graph $G$ is $\left(a,b\right)$-choosable if given any list assignment $L$ with $\text{◂=▸}\mid L\left(v\right)\mid =a$ for each $\text{◂+▸}v\in V\left(G\right)$ there exists a function $\phi$ such that $\text{◂⊆▸}\phi \left(v\right)\subseteq L\left(v\right)$ and $\text{◂=▸}\mid \phi \left(v\right)\mid =b$ for all $\text{◂+▸}v\in V\left(G\right)$, and whenever vertices $x$ and $y$ are adjacent $\text{◂+▸}\phi \left(x\right)\cap \phi \left(y\right)=\varnothing$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture.     

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 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 $m$.