arXiv subjects

Daniel Soltész

Publications and source records attributed to Daniel Soltész.

7 recordsLinked to original sources

arXiv2019

New bounds on even cycle creating Hamiltonian paths using expander graphs

We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and Körner proved \[n^{\frac{1}{2}n-o(n)}\leq H_n(C_4) \leq n^{\frac{3}{4}n+o(n)}.\] In this paper we close the superexponential gap between their lower and upper bounds by proving \[n^{\frac{1}{2}n-\frac{1}{2}\frac{n}{\log{n}}-O(1)}\leq H_n(C_4) \leq n^{\frac{1}{2}n+o\left(\frac{n}{\log{n}} \right)}.\] We also improve the previously established upper bounds on $H_n(C_{2k})$ for $k>3$, and we present a small improvement on the lower bound of Füredi, Kantor, Monti and Sinaimeri on the maximum number of so-called pairwise reversing permutations. One of our main tools is a theorem of Krivelevich, which roughly states that (certain kinds of) good expanders contain many Hamiltonian paths.

math.CO↗

arXiv2019

Questions on the Structure of Perfect Matchings inspired by Quantum Physics

We state a number of related questions on the structure of perfect matchings. Those questions are inspired by and directly connected to Quantum Physics. In particular, they concern the constructability of general quantum states using modern photonic technology. For that we introduce a new concept, denoted as inherited vertex coloring. It is a vertex coloring for every perfect matching. The colors are inherited from the color of the incident edge for each perfect matching. First, we formulate the concepts and questions in pure graph-theoretical language, and finally we explain the physical context of every mathematical object that we use. Importantly, every progress towards answering these questions can directly be translated into new understanding in quantum physics.

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arXiv2018

Even cycle creating paths

We say that two graphs $H_1,H_2$ on the same vertex set are $G$-creating ($G$-different in other papers, this difference is explained in the introduction) if the union of the two graphs contains $G$ as a subgraph. Let $H(n,k)$ be the maximal number of pairwise $C_k$-creating paths (of arbitrary length) on $n$ vertices. The behaviour of $H(n,2k+1)$ is much better understood than the behaviour of $H(n,2k)$, the former is an exponential function of $n$ while the latter is larger than exponential, for every fixed $k$. We study $H(n,k)$ for fixed $k$ and $n$ tending to infinity. The only non trivial upper bound on $H(n,2k)$ was in the case where $k=2$ $$H(n,4)\leq n^{\left(1-\frac{1}{4} \right) n-o(n)}, $$ this was proved by Cohen, Fachini and Körner. In this paper, we generalize their method to prove that for every $k \geq 2$, $$H(n,2k) \leq n^{\left( 1- \frac{2}{3k^2-2k} \right)n-o(n)}. $$ Our proof uses constructions of bipartite, regular, $C_{2k}$-free graphs with many edges by Reiman, Benson, Lazebnik, Ustimenko and Woldar. For some special values of $k$ we can have slightly denser such bipartite graphs than for general $k$, this results in having better upper bounds on $H(n,2k)$ than stated above for these special values of $k$.

math.CO↗

arXiv2017

On k-neighbor separated permutations

Two permutations of $[n]=\{1,2 \ldots n\}$ are \textit{$k$-neighbor separated} if there are two elements that are neighbors in one of the permutations and that are separated by exactly $k-2$ other elements in the other permutation. Let the maximal number of pairwise $k$-neighbor separated permutations of $[n]$ be denoted by $P(n,k)$. In a previous paper, the authors have determined $P(n,3)$ for every $n$, answering a question of Körner, Messuti and Simonyi affirmatively. In this paper we prove that for every fixed positive integer $\ell $, $$P(n,2^\ell+1) = 2^{n-o(n)}. $$ We conjecture that for every fixed even $k$, $P(n,k)=2^{n-o(n)}$. We also show that this conjecture is asymptotically true in the following sense $$\lim_{k \rightarrow \infty} \lim_{n \rightarrow \infty} \sqrt[n]{P(n,k)}=2.$$ Finally, we show that for even $n$, $P(n,n)= 3n/2$.

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arXiv2016

Independent sets in the union of two Hamiltonian cycles

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c 12$ vertices is the union of two Hamiltonian cycles and $α(G)=n/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.

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arXiv2015

New bounds on Simonyi's conjecture

We say that a pair $(\mathcal{A},\mathcal{B})$ is a recovering pair if $\mathcal{A}$ and $\mathcal{B}$ are set systems on an $n$ element ground set, such that for every $A,A' \in \mathcal{A}$ and $B,B' \in \mathcal{B}$ we have that ($A \setminus B = A' \setminus B'$ implies $A=A'$) and symmetrically ($B \setminus A = B' \setminus A'$ implies $B=B'$). G. Simonyi conjectured that if $(\mathcal{A},\mathcal{B})$ is a recovering pair, then $|\mathcal{A}||\mathcal{B}|\leq 2^n$. For the quantity $|\mathcal{A}||\mathcal{B}|$ the best known upper bound is $2.3264^n$ due to Körner and Holzman. In this paper we improve this upper bound to $2.284^n$. Our proof is combinatorial.

math.CO↗

arXiv2015

On the 1-switch conjecture in the Hypercube and other graphs

Feder and Subi conjectured that for any $2$-coloring of the edges of the $n$-dimensional cube, we can find an antipodal pair of vertices connected by a path that changes color at most once. We discuss the case of random colorings, and we prove the conjecture for a wide class of colorings. Our method can be applied to a more general problem, where $Q_n$ can be replaced by any graph $G$, the notion of antipodality by a fixed automorphism $ϕ\in Aut(G)$. Thus for any $2$-coloring of $E(G)$ we are looking for a pair of vertices $u,v$ such that $u= ϕ(v)$ and there is a path between them with as few color changes as possible. We solve this problem for the toroidal grid $G=C_{2a} \square c_{2b}$ with the automorphism that takes every vertex to its unique farthest pair. Our results point towards a more general conjecture which turns out to be supported by a previous theorem of Feder and Subi.

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