Path coverings with prescribed ends in faulty hypercubes

We discuss the existence of vertex disjoint path coverings with prescribed ends for the $n$-dimensional hypercube with or without deleted vertices. Depending on the type of the set of deleted vertices and desired properties of the path coverings we establish the minimal integer $m$ such that for every $n \ge m$ such path coverings exist. Using some of these results, for $k \le 4$, we prove Locke's conjecture that a hypercube with $k$ deleted vertices of each parity is Hamiltonian if $n \ge k +2.$ Some of our lemmas substantially generalize known results of I. Havel and T. Dvo\v{r}\'{a}k. At the end of the paper we formulate some conjectures supported by our results.Comment: 26 page

Parity balance of the ii-th dimension edges in Hamiltonian cycles of the hypercube

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in $Q_n$ contains two opposite edges in a 4-cycle. We prove this conjecture for $n \le 7$, and for any Hamiltonian cycle containing more than $2^{n-2}$ edges in the same dimension. This bound is finally improved considering the equi-independence number of $Q_{n-1}$, which is a concept introduced in this paper for bipartite graphs

On realization graphs of degree sequences

Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.Comment: 10 pages, 5 figure

Extending a perfect matching to a Hamiltonian cycle

Graph TheoryInternational audienceRuskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property

Geodesic bipancyclicity of the Cartesian product of graphs

A cycle containing a shortest path between two vertices $u$ and $v$ in a graph $G$ is called a $(u,v)$-geodesic cycle. A connected graph $G$ is geodesic 2-bipancyclic, if every pair of vertices $u,v$ of it is contained in a $(u,v)$-geodesic cycle of length $l$ for each even integer $l$ satisfying $2d + 2\leq l \leq |V(G)|,$ where $d$ is the distance between $u$ and $v.$ In this paper, we prove that the Cartesian product of two geodesic hamiltonian graphs is a geodesic 2-bipancyclic graph. As a consequence, we show that for $n \geq 2$ every $n$-dimensional torus is a geodesic 2-bipancyclic graph

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.Comment: 14 pages, 4 figure