相关论文: The proofs of two directed paths conjectures of Bo…

The conjecture of Beineke and Harary states that for any two vertices which can be separated by $k$ vertices and $l$ edges for $l\geq 1$ but neither by $k$ vertices and $l-1$ edges nor $k-1$ vertices and $l$ edges there are $k+l$…

Menger's well-known theorem from 1927 characterizes when it is possible to find $k$ vertex-disjoint paths between two sets of vertices in a graph $G$. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs,…

We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree…

The hypercube \( Q_n \) contains a Hamiltonian path joining \( x \) and \( y \) (where $x$ and $y$ from the opposite partite set) containing \( P \) if and only if the induced subgraph of \( P \) is a linear forest, where none of these…

The balanced hypercube $BH_n$, a variant of the hypercube, was proposed as a desired interconnection network topology. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2,\cdots,s_{2n-2}\}$ and $T=\{t_1,t_2,\cdots,t_{2n-2}\}$ are…

We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ into graphs from various classes, the simplest cases of paths…

Motivated by applications to monotonicity testing, Lehman and Ron (JCTA, 2001) proved the existence of a collection of vertex disjoint paths between comparable sub-level sets in the directed hypercube. The main technical contribution of…

As a variant of the well-known hypercube, the balanced hypercube $BH_n$ was proposed as a novel interconnection network topology for parallel computing. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2\}$ and $T=\{t_1,t_2\}$…

In the year 1990, B\'ela Bollob\'as, Imre Leader and Andrew Radcliffe considered the following combinatorial problem: given three parameters k, n and q, find a set of k vertices in the binary n-cube which contains a maximal number of…

A conjecture attributed to Smith states that every pair of longest cycles in a $k$-connected graph intersect each other in at least $k$ vertices. In this paper, we show that every pair of longest cycles in a~$k$-connected graph on $n$…

Stein (2020) conjectured that for any positive integer $k$, every oriented graph of minimum semi-degree greater than $k/2$ contains every oriented path of length $k$. This conjecture is true for directed paths by a result from Jackson (JGT,…

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s,t and k edge-disjoint s-t paths, also contains two vertices u,v and f(k) independent u-v paths. Here, a set of paths is independent if none of…

A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple…

Two $a{-}b$ paths in a graph $G$ are order-compatible if their common vertices occur in the same order when travelling from $a$ to $b$. Suppose a graph contains an infinite number $\delta$ of edge-disjoint $a{-}b$ paths. G.A. Dirac asked…

We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges.…

We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture,…

This paper considers the problem of many-to-many disjoint paths in the hypercube $Q_n$ with $f$ faulty vertices and obtains the following result. For any integer $k$ with $1\leq k\leq n-2$, any two sets $S$ and $T$ of $k$ fault-free…

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…

Two graphs $G_1$ and $G_2$ on $n$ vertices are said to \textit{pack} if there exist injective mappings of their vertex sets into $[n]$ such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollob\'as and…

The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional…