arXiv subjects

Tomáš Dvořák

Publications and source records attributed to Tomáš Dvořák.

5 recordsLinked to original sources

arXiv2023

Paired 2-disjoint path covers of burnt pancake graphs with faulty elements

The burnt pancake graph $BP_n$ is the Cayley graph of the hyperoctahedral group using prefix reversals as generators. Let $\{u,v\}$ and $\{x,y\}$ be any two pairs of distinct vertices of $BP_n$ for $n\geq 4$. We show that there are $u-v$ and $x-y$ paths whose vertices partition the vertex set of $BP_n$ even if $BP_n$ has up to $n-4$ faulty elements. On the other hand, for every $n\ge3$ there is a set of $n-2$ faulty edges or faulty vertices for which such a fault-free disjoint path cover does not exist.

cs.DM↗

arXiv2019

Neighbor connectivity of $k$-ary $n$-cubes

The neighbor connectivity of a graph $G$ is the least number of vertices such that removing their closed neighborhoods from $G$ results in a graph that is disconnected, complete or empty. If a~graph is used to model the topology of an interconnection network, this means that the failure of a network node causes failures of all its neighbors. We completely determine the neighbor connectivity of $k$-ary $n$-cubes for all $n\ge1$ and $k\ge2$.

cs.DM↗

arXiv2017

Generalized Gray codes with prescribed ends of small dimensions

Given pairwise distinct vertices $\{α_i , β_i\}^k_{i=1}$ of the $n$-dimensional hypercube $Q_n$ such that the distance of $α_i$ and $β_i$ is odd, are there paths $P_i$ between $α_i$ and $β_i$ such that $\{V (P_i)\}^k_{i=1}$ partitions $V(Q_n)$? A positive solution for every $n\ge1$ and $k=1$ is known as a Gray code of dimension $n$. In this paper we settle this problem for small values of $n$.

cs.DM↗

arXiv2016

Matchings of quadratic size extend to long cycles in hypercubes

Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size. In this paper we show that there is a quadratic function $q(n)$ such that every matching in the $n$-dimensional hypercube of size at most $q(n)$ may be extended to a cycle which covers at least $\frac34$ of the vertices.

cs.DM↗

arXiv2016

Generalized Gray codes with prescribed ends

An $n$-bit Gray code is a sequence of all $n$-bit strings such that consecutive strings differ in a single bit. It is well-known that given $α,β\in\{0,1\}^n$, an $n$-bit Gray code between $α$ and $β$ exists iff the Hamming distance $d(α,β)$ of $α$ and $β$ is odd. We generalize this classical result to $k$ pairwise disjoint pairs $α_i, β_i\in\{0,1\}^n$: if $d(α_i,β_i)$ is odd for all $i$ and $k 1$ with one exception in the case when $n = k + 1 = 4$. Our result is optimal in the sense that for every $n>2$ there are $n$ pairwise disjoint pairs $α_i,β_i\in\{0,1\}^n$ with $d(α_i,β_i)$ odd for which such sequences do not exist.

cs.DM↗