arXiv subjects
Publications and source records attributed to Charles Kenney.
A random set $S$ is $p$-spread if, for all sets $T$, $$\mathbb{P}(S \supseteq T) \leq p^{|T|}.$$ There is a constant $C>1$ large enough that for every graph $G$ with maximum degree $D$, there is a $C/D$-spread distribution on $(D+1)$-colorings of $G$. Making use of a connection between thresholds and spread distributions due to Frankston, Kahn, Narayanan, and Park, a palette sparsification theorem of Assadi, Chen, and Khanna follows.
It is shown that the following holds for each $\varepsilon >0$. For $G$ an $n$-vertex graph of maximum degree $D$, lists $S_v$ of size $D+1$ (for $v\in V(G)$), and $L_v$ chosen uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $S_v$ (independent of other choices), \[ \mbox{$G$ admits a proper coloring $\sigma$ with $\sigma_v\in L_v$ $\forall v$} \] with probability tending to 1 as $D\to \infty$. When each $S_v $ is $\{1\dots D+1\}$, this is an asymptotically optimal version of the ``palette sparsification'' theorem of Assadi, Chen and Khanna that was proved in an earlier paper by the present authors.
Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient polynomial $p(x, y, z)$ whose only integer roots are consecutive triples of Tribonacci numbers.
It is shown that the following holds for each $\varepsilon>0$. For $G$ an $n$-vertex graph of maximum degree $D$ and "lists" $L_v$ ($v \in V(G)$) chosen independently and uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $\{1, ..., D+1\}$, \[ G \text{ admits a proper coloring } \sigma \text{ with } \sigma_v \in L_v \forall v \] with probability tending to 1 as $D \to \infty$. This is an asymptotically optimal version of a recent "palette sparsification" theorem of Assadi, Chen, and Khanna.
Given a graph with a designated set of boundary vertices, we define a new notion of a Neumann Laplace operator on a graph using a reflection principle. We show that the first eigenvalue of this Neumann graph Laplacian satisfies a Cheeger inequality.