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Liana Yepremyan

Publications and source records attributed to Liana Yepremyan.

45 records · Page 3Linked to original sources
arXiv2017

Rainbow matchings in properly-coloured multigraphs

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by $n$ colours with at least $n + 1$ edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities $o(n)$ that is properly edge-coloured by $n$ colours with at least $n + o(n)$ edges of each colour there must be a matching of size $n-O(1)$ that uses each colour at most once.

math.CO↗
arXiv2017

Supersaturation of Even Linear Cycles in Linear Hypergraphs

A classic result of Erdős and, independently, of Bondy and Simonovits says that the maximum number of edges in an $n$-vertex graph not containing $C_{2k}$, the cycle of length $2k$, is $O( n^{1+1/k})$. Simonovits established a corresponding supersaturation result for $C_{2k}$'s, showing that there exist positive constants $C,c$ depending only on $k$ such that every $n$-vertex graph $G$ with $e(G)\geq Cn^{1+1/k}$ contains at least $c\left(\frac{e(G)}{v(G)}\right)^{2k}$ many copies of $C_{2k}$, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper, we extend Simonovits' result to a supersaturation result of $r$-uniform linear cycles of even length in $r$-uniform linear hypergraphs. Our proof is self-contained and includes the $r=2$ case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.

math.CO↗
arXiv2017

A Turán theorem for extensions via an Erdős-Ko-Rado theorem for Lagrangians

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine the Turán number of the extension of an $r$-graph consisting of two vertex-disjoint edges, settling a conjecture of Hefetz and Keevash, who previously determined this Turán number for $r=3$. As the key ingredient of the proof we show that the Lagrangian of intersecting $r$-graphs is maximized by principally intersecting $r$-graphs for $r \geq 4$.

math.CO↗
arXiv2015

Complete graph immersions and minimum degree

An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths P_{uv} are internally disjoint from phi(V(H)). We prove that every simple graph of minimum degree at least 11t+7 contains a strong immersion of the complete graph K_t. This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al.

math.CO↗
arXiv2015

Turán numbers of extensions

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $r-2$ new vertices. Keevash and Sidorenko~ have previously determined Turán densities of two families of hypergraph extensions. We determine the Turán numbers for these families, using classical stability techniques and new tools introduced in our earlier paper.

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arXiv2015

The extremal function for disconnected minors

For a graph $H$ let $c(H)$ denote the supremum of $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $$c(H) \leq \frac{|V(H)|+\mathrm{comp}(H)}{2}-1,$$ when $H$ is a union of cycles, verifying conjectures of Reed and Wood, and Harvey and Wood. We derive the above result from a theorem which allows us to find two vertex disjoint subgraphs with prescribed densities in a sufficiently dense graph, which might be of independent interest.

math.CO↗
arXiv2015

Turán Number of Generalized Triangles

The family $Σ_r$ consists of all $r$-graphs with three edges $D_1,D_2,D_3$ such that $|D_1\cap D_2|=r-1$ and $D_1 \triangle D_2 \subseteq D_3$. A generalized triangle, $\mathcal{T}_r \in Σ_r$ is an $r$-graph on $\{1,2,\ldots,2r-1\}$ with three edges $D_1, D_2, D_3$, such that $D_1=\{1,2,\dots,r-1, r\}, D_2= \{1, 2, \dots, r-1, r+1 \}$ and $D_3 = \{r, r+1, \dots, 2r-1\}.$ Frankl and Füredi conjectured that for all $r\geq 4$, $ex(n,Σ_r) = ex(n,\mathcal{T}_r )$ for all sufficiently large $n$ and they also proved it for $r=3$. Later, Pikhurko showed that the conjecture holds for $r=4$. In this paper we determine $ex(n,\mathcal{T}_5)$ and $ex(n,\mathcal{T}_6)$ for sufficiently large $n$, proving the conjecture for $r=5,6$.

math.CO↗
arXiv2015

Sparse halves in dense triangle-free graphs

Erdős conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least $\frac{2}{5}n$. Keevash and Sudakov improved this result to graphs with average degree at least $\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\frac{5}{14}n$ and for graphs with average degree at least $(\frac{2}{5} - \varepsilon)n$ for some absolute $\varepsilon >0$. Moreover, we show that the conjecture is true for graphs which are close to the Petersen graph in edit distance.

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