arXiv subjects

Micha Christoph

Publications and source records attributed to Micha Christoph.

22 records · Page 2Linked to original sources

arXiv2024

The power of many colours

A classical problem, due to Gerencsér and Gyárfás from 1967, asks how large a monochromatic connected component can we guarantee in any $r$-edge colouring of $K_n$? We consider how big a connected component can we guarantee in any $r$-edge colouring of $K_n$ if we allow ourselves to use up to $s$ colours. This is actually an instance of a more general question of Bollobás from about 20 years ago which asks for a $k$-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided $n$ is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form. We also consider a generalisation in a similar direction of a question first considered by Erdős and Rényi in 1956, who considered given $n$ and $m$, what is the smallest number of $m$-cliques which can cover all edges of $K_n$? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any $r$-edge colouring of $K_n$. We consider what happens if we allow ourselves to use up to $s$ colours. We obtain a more complete understanding of the answer to this question for large $n$, in particular determining it up to a constant factor for all $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.

math.CO↗

arXiv2024

The Hamilton space of pseudorandom graphs

We show that if $n$ is odd and $p \ge C \log n / n$, then with high probability Hamilton cycles in $G(n,p)$ span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph $G$, that is, a graph $G$ with odd $n$ vertices and minimum degree $n/2 + C$ for sufficiently large constant $C$, span its cycle space.

math.CO↗

arXiv2023

A note on digraph splitting

A tantalizing open problem, posed independently by Stiebitz in 1995 and by Alon in 2006, asks whether for every pair of integers $s,t \ge 1$ there exists a finite number $F(s,t)$ such that the vertex set of every digraph of minimum out-degree at least $F(s,t)$ can be partitioned into non-empty parts $A$ and $B$ such that the subdigraphs induced on $A$ and $B$ have minimum out-degree at least $s$ and $t$, respectively. In this short note, we prove that if $F(2,2)$ exists, then all the numbers $F(s,t)$ with $s,t\ge 1$ exist and satisfy $F(s,t)=Θ(s+t)$. In consequence, the problem of Alon and Stiebitz reduces to the case $s=t=2$. Moreover, the numbers $F(s,t)$ with $s,t \ge 2$ either all exist and grow linearly, or all of them do not exist.

math.CO↗

arXiv2023

Minimum Degree Threshold for $H$-factors with High Discrepancy

Given a graph $H$, a perfect $H$-factor in a graph $G$ is a collection of vertex-disjoint copies of $H$ spanning $G$. Kühn and Osthus showed that the minimum degree threshold for a graph $G$ to contain a perfect $H$-factor is either given by $1-1/χ(H)$ or by $1-1/χ_{cr}(H)$ depending on certain natural divisibility considerations. Given a graph $G$ of order $n$, a $2$-edge-coloring of $G$ and a subgraph $G'$ of $G$, we say that $G'$ has high discrepancy if it contains significantly (linear in $n$) more edges of one color than the other. Balogh, Csaba, Pluhár and Treglown asked for the minimum degree threshold guaranteeing that every 2-edge-coloring of $G$ has an $H$-factor with high discrepancy and they settled the case where $H$ is a clique. Here we completely resolve this question by determining the minimum degree threshold for high discrepancy of $H$-factors for every graph $H$.

math.CO↗