arXiv subjects

Shoham Letzter

Publications and source records attributed to Shoham Letzter.

59 records · Page 4Linked to original sources

arXiv2015

Monochromatic cycle partitions of $2$-coloured graphs with minimum degree $3n/4$

Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour. We prove that this conjecture holds for $n$ large enough, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.

math.CO↗

arXiv2014

Path Ramsey number for random graphs

Answering a question raised by Dudek and Prałat, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that $2/3$ cannot be replaced by a larger constant. As part of the proof we obtain the following result which may be of independent interest. We show that given a graph $G$ on $n$ vertices with at least $(1-ε)\binom{n}{2}$ edges, whenever $G$ is $2$-edge-coloured, there is a monochromatic path of length at least $(2/3-100\sqrtε)n$. This is an extension of the classical result by Gerencsér and Gyárfás which says that whenever $K_n$ is $2$-coloured there is a monochromatic path of length at least $2n/3$.

math.CO↗

arXiv2013

Large monochromatic triple stars in edge colourings

Following problems posed by Gyárfás, we show that for every $r$-edge-colouring of $K_n$ there is a monochromatic triple star of order at least $n/(r-1)$, improving a previous result by Ruszinkó. An edge colouring of a graph is called a local $r$-colouring if every vertex spans edges of at most $r$ distinct colours. We prove the existence of a monochromatic triple star with at least $rn/(r^2-r+1)$ vertices in every local $r$-colouring of $K_n$.

math.CO↗

arXiv2013

The Property of Having a $k$-Regular Subgraph Has a Sharp Threshold

We prove that the property of containing a $k$-regular subgraph in the random graph model $G(n,p)$ has a sharp threshold for $k\ge3$. We also show how to use similar methods to obtain an easy prove for the (known fact of) sharpness of having a non empty $k$-core for $k\ge3$.

math.PR↗

arXiv2013

Radon Numbers for Trees

Many interesting problems are obtained by attempting to generalize classical results on convexity in Euclidean spaces to other convexity spaces, in particular to convexity spaces on graphs. In this paper we consider $P_3$-convexity on graphs. A set $U$ of vertices in a graph $G$ is $P_3$-convex if every vertex not in $U$ has at most one neighbour in $U$. More specifically, we consider Radon numbers for $P_3$-convexity in trees. Tverberg's theorem states that every set of $(k-1)(d+1)-1$ points in $\mathbb{R}^d$ can be partitioned into $k$ sets with intersecting convex hulls. As a special case of Eckhoff's conjecture, we show that a similar result holds for $P_3$-convexity in trees. A set $U$ of vertices in a graph $G$ is called free, if no vertex of $G$ has more than one neighbour in $U$. We prove an inequality relating the Radon number for $P_3$-convexity in trees with the size of a maximal free set.

math.CO↗