arXiv subjects

Lenka Kopfová

Publications and source records attributed to Lenka Kopfová.

4 recordsLinked to original sources

arXiv2026

On the probability that a random Latin square is all-even

In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\times n$ Latin square, the probability that all rows are even is of the form $2^{-n+o(n)}$. This fills a gap left by recent work of Kwan, Petrova and Sawhney: they proved that the row parities behave like a sequence of independent coin flips in a variety of different senses, but they were not able to lower-bound the all-even probability, as their approach had a small number of "exceptional rows" whose parities they could not control. In fact, we prove a more general theorem on the probability of any possible prescription of row parities, column parities and symbol parities. Our approach extends the ideas of Kwan, Petrova and Sawhney, introducing a new way to "bootstrap" intercalate switchings to control more complex switching operations.

math.CO↗

arXiv2026

Proof of the mad conjecture and its coloring applications

For a finite graph $G$, the maximum average degree $\operatorname{mad}(G)$ is the largest average degree of a nonempty subgraph of $G$. Hendrey, Norin and Wood asked whether this parameter is partitionable, that is, whether for all positive reals $a,b$ every graph $G$ with $\operatorname{mad}(G)<a+b$ admits a vertex partition $V(G)=A\cup B$ with $\operatorname{mad}(G[A])<a$ and $\operatorname{mad}(G[B])<b$. In this note, we answer this question affirmatively. Moreover, we generalize this to an arbitrary number of partition classes. The clustered chromatic number $χ_\star(\mathrm{G})$ of a graph class $\mathrm{G}$ is the minimum integer $k$ such that, for some integer $c$, every graph in $\mathrm{G}$ has a $k$-coloring in which every monochromatic component has at most $c$ vertices. We apply this result to show that $χ_\star(\mathrm{A}_m)=\left\lfloor\frac{m}{2}\right\rfloor+1$, where $\mathrm{A}_m$ is the family of graphs $G$ with $\operatorname{mad}(G)\leq m$. This solves an open problem posed in Wood's survey and highlighted by Hendrey and Wood. As another application of our general partition result for $\operatorname{mad}$, we obtain a bound for relaxed colorings. Namely, for non-negative integers $d_1,\ldots,d_k$, every graph $G$ with \[\operatorname{mad}(G)<\sum_{i=1}^k \frac{2d_i+2}{d_i+2}\] admits a partition $V(G)=V_1\cup\ldots\cup V_k$ such that $Δ(G[V_i])\leq d_i$ for each $i\in[k]$. In particular, this provides the first non-trivial bounds for $k\ge3$ with arbitrary $d_i$ and improves the previously known bound for $(d+1,d)$-colorings with $d \ge 2$.

math.CO↗

arXiv2026

Replacers and their evolutionary stability in the Moran process on graphs

Evolutionary dynamics in finite structured populations are commonly modeled by the Moran Birth-death process. A key quantity is the fixation probability of a single invader attempting to take over a population of residents. A recent work introduced a new neighborhood-aware phenotype called a replacer. A replacer never wastes their reproductive turn by always replacing an individual of the other type (if available). In this work, we study the evolutionary stability of resident replacers who are invaded by mutant replacers. We find that residents are strongly protected against such invasions, and we quantify the strength of this effect by showing three types of results. First, we show that on well-mixed populations of size $N$, the invader fixation probability is exponentially small in $N$, even when the invader has a fixed relative reproductive rate $r>1$, and the same holds for all high-degree graphs. Second, we study bounded-degree graphs. We prove that on cycles, the fixation probability of an advantageous invader decreases only by a constant factor. However, we also present graphs with maximum degree 4, where the invader fixation probability is exponentially small in $N$ whenever $r\le1.9$. Thus, high degrees are sufficient for evolutionary stability, whereas with low degrees the evolutionary stability depends on specific features of the underlying spatial structure. Third, we prove general bounds for arbitrary graphs. Namely, we show that for any graph $G$ the invader fixation probability drops below the natural baseline given by the standard Moran process with oblivious individuals on a well-mixed population, both for $r\approx 1$ and for $r\ge 2$. Together, our results establish that the evolutionary dynamics of replacers is better characterized by the phrase ``survival of the first'' rather than the classic ``survival of the fittest''.

q-bio.PE↗

arXiv2024

Colonization times in Moran process on graphs

Moran Birth-death process is a standard stochastic process that is used to model natural selection in spatially structured populations. A newly occurring mutation that invades a population of residents can either fixate on the whole population or it can go extinct due to random drift. The duration of the process depends not only on the total population size $n$, but also on the spatial structure of the population. In this work, we consider the Moran process with a single type of individuals who invade and colonize an otherwise empty environment. Mathematically, this corresponds to the setting where the residents have zero reproduction rate, thus they never reproduce. We present two main contributions. First, in contrast to the Moran process in which residents do reproduce, we show that the colonization time is always at most a polynomial function of the population size $n$. Namely, we show that colonization always takes at most $\frac12n^3-\frac12n^2$ expected steps, and for each $n$, we exactly identify the unique slowest spatial structure where it takes exactly that many steps. Moreover, we establish a stronger bound of roughly $n^{2.5}$ steps for spatial structures that contain only two-way connections and an even stronger bound of roughly $n^2$ steps for lattice-like spatial structures. Second, we discuss various complications that one faces when attempting to measure fixation times and colonization times in spatially structured populations, and we propose to measure the real duration of the process, rather than counting the steps of the classic Moran process.

q-bio.PE↗