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Maya Stein

Publications and source records attributed to Maya Stein.

At least 55 records · Page 3Linked to original sources
arXiv2018

Degree conditions for embedding trees

We conjecture that every $n$-vertex graph of minimum degree at least $\frac k2$ and maximum degree at least $2k$ contains all trees with $k$ edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our work also has implications on the Erd\H os--Sós conjecture and the $\frac 23$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.

math.CO↗
arXiv2018

Monochromatic tree covers and Ramsey numbers for set-coloured graphs

We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-uniform hypergraphs: they have a transversal of size at most $r-k$. (Similar results have been obtained by Király et al., see below.) However, we also show that the bound $r-k$ is not best possible in general.

math.CO↗
arXiv2018

Partitioning $2$-coloured complete $k$-uniform hypergraphs into monochromatic $\ell$-cycles

We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$ vertices has at most two disjoint monochromatic $\ell$-cycles in different colours that together cover all but at most $4(k-\ell)$ vertices. If $\ell \leq k/3$, then at most two $\ell$-cycles cover all but at most $2(k-\ell)$ vertices. Furthermore, we can cover all vertices with at most $4$ ($3$ if $\ell\leq k/3$) disjoint monochromatic $\ell$-cycles.

math.CO↗
arXiv2018

The domino problem on groups of polynomial growth

We characterize the virtually nilpotent finitely generated groups (or, equivalently by Gromov's theorem, groups of polynomial growth) for which the Domino Problem is decidable: These are the virtually free groups, i.e. finite groups, and those having $\Z$ as a subgroup of finite index.

math.GR↗
arXiv2017

Clique coloring $B_1$-EPG graphs

We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are $2$-clique colorable. In this paper we prove that $B_1$-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are $4$-clique colorable. Moreover, given a $B_1$-EPG representation of a graph, we provide a linear time algorithm that constructs a $4$-clique coloring of it.

math.CO↗
arXiv2016

Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.

math.CO↗
arXiv2016

Approximately coloring graphs without long induced paths

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.

math.CO↗
arXiv2015

Convex p-partitions of bipartite graphs

A set of vertices X of a graph G is convex if it contains all vertices on shortest paths between vertices of X. We prove that for fixed p, all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial time.

math.CO↗
arXiv2015

The approximate Loebl-Komlós-Sós Conjecture

We prove the following version of the Loebl-Komlos-Sos Conjecture: For every alpha>0 there exists a number M such that for every k>M every n-vertex graph G with at least (0.5+alpha)n vertices of degree at least (1+alpha)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy common to approaches which employ the Szemeredi Regularity Lemma: we decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. However, the decomposition given by the Regularity Lemma is not of help when G is sparse. To surmount this shortcoming we use a more general decomposition technique: each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the Regularity Lemma), and two other objects each exhibiting certain expansion properties.

math.CO↗
arXiv2015

Monochromatic cycle partitions in local edge colourings

An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number $r$, we show that the vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r^2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.

math.CO↗
arXiv2015

The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs

Loebl, Komlós and Sós conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. For our proof, we use a structural decomposition which can be seen as an analogue of Szemerédi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].

math.CO↗
arXiv2014

Partitioning two-coloured complete multipartite graphs into monochromatic paths and cycles

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded. From this we derive that the whole graph, if large enough, may be covered with 14 vertex-disjoint monochromatic cycles.

math.CO↗
arXiv2014

Connectivity and tree structure in finite graphs

Considering systems of separations in a graph that separate every pair of a given set of vertex sets that are themselves not separated by these separations, we determine conditions under which such a separation system contains a nested subsystem that still separates those sets and is invariant under the automorphisms of the graph. As an application, we show that the $k$-blocks -- the maximal vertex sets that cannot be separated by at most $k$ vertices -- of a graph $G$ live in distinct parts of a suitable tree-decomposition of $G$ of adhesion at most $k$, whose decomposition tree is invariant under the automorphisms of $G$. This extends recent work of Dunwoody and Krön and, like theirs, generalizes a similar theorem of Tutte for $k=2$. Under mild additional assumptions, which are necessary, our decompositions can be combined into one overall tree-decomposition that distinguishes, for all $k$ simultaneously, all the $k$-blocks of a finite graph.

math.CO↗
arXiv2014

Linear and projective boundaries in HNN-extensions and distortion phenomena

Linear and projective boundaries of Cayley graphs were introduced in~\cite{kst} as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits $g^\infty=\{g^i: i\in \mathbb N\}$, or orbits $g^{\pm\infty}=\{g^i:i\in\mathbb Z\}$, respectively, of non-torsion elements~$g$ of the group $G$, where `sufficiently close' (forward) orbits become identified, together with a metric bounded by 1. We show that for all finitely generated groups, the distance between the antipodal points $g^\infty$ and $g^{-\infty}$ in the linear boundary is bounded from below by $\sqrt{1/2}$, and we give an example of a group which has two antipodal elements of distance at most $\sqrt{12/17}<1$. Our example is a derivation of the Baumslag-Gersten group. \newline We also exhibit a group with elements $g$ and $h$ such that $g^\infty = h^\infty$, but $g^{-\infty}\neq h^{-\infty}$. Furthermore, we introduce a notion of average-case-distortion---called growth---and compute explicit positive lower bounds for distances between points $g^\infty$ and $h^\infty$ which are limits of group elements $g$ and $h$ with different growth.

math.GR↗
arXiv2014

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χ_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ(G),\ldots,χ_b(G), and b-monotonic if χ_b(H_1) \geq χ_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.

cs.CC↗
arXiv2013

List edge-colouring and total colouring in graphs of low treewidth

We prove that the list chromatic index of a graph of maximum degree $Δ$ and treewidth $\leq \sqrt{2Δ} -3$ is $Δ$; and that the total chromatic number of a graph of maximum degree $Δ$ and treewidth $\leq Δ/3 +1$ is $Δ+1$. This improves results by Meeks and Scott.

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