Related papers: Paired many-to-many 2-disjoint path cover of Johns…

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…

A paired $k$-to-$k$ disjoint path cover of a graph $G$ is a collection of pairwise disjoint path subgraphs $P_1,P_2,\dotsc,P_k$ such that each $P_i$ has prescribed vertices $s_i$ and $t_i$ as endpoints and the union of $P_1,P_2,\dotsc,P_k$…

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…

A path cover of a graph is a set of disjoint paths so that every vertex in the graph is contained in one of the paths. The path cover number $p(G)$ of graph $G$ is the cardinality of a path cover with the minimum number of paths. Reed in…

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 $T=\{t_1,t_2\}$…

Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic…

A graph $G$ is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. It is well known that a 2-connected, $k$-regular graph $G$ on at most $3k-1$ vertices is edge-Hamiltonian if for every edge…

For any given $n,m \in \mathbb{N}$ with $ m < n $, the Johnson graph $J(n,m)$ is defined as the graph whose vertex set is $V=\{v\mid v\subseteq [n]=\{1,...,n\}, |v|=m\}$, where two vertices $v$,$w$ are adjacent if and only if $|v\cap…

Let $B$ be a bidirected multigraph with signing $\sigma$, let $X$ be a set of vertices in $B$, and let $k$ be a non-negative integer. For any pair of vertex sets $S,T\subset V(B)$ satisfying $X\cap S = X\cap T$, we denote by $B_{S,T}$ the…

Two plane drawings of graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common. Let $S$ be a convex point set of $2n \geq 10$ points and let $\mathcal{H}$ be a family of…

The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the…

We give a construction that provides infinitely many 2-connected, cubic, bipartite, and planar graphs G with 3k vertices and such that the number of disjoint copies of a 3-vertex path in G is less than k.

Let $H$ be an $n$-vertex 3-uniform hypergraph such that every pair of vertices is in at least $n/3+o(n)$ edges. We show that $H$ contains two vertex-disjoint tight paths whose union covers the vertex set of $H$. The quantity two here is…

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord\'an [4] and a theorem of Jord\'an [6] on packing of rigid spanning…

A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected…

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 $u-v$…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, a path in $G$ is said to be an $S$-path if it connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E(P_1)\cap…

The balanced hypercube $BH_{n}$, proposed by Wu and Huang, is a variation of the hypercube. The paired 1-disjoint path cover of $BH_{n}$ is the Hamiltonian laceability, which was obtained by Xu et al. in [Appl. Math. Comput. 189 (2007)…

We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is…

A balanced bipartite graph $G$ is said to be $2p$-Hamilton-biconnected if for any balanced subset $W$ of size $2p$ of $V(G)$, the subgraph induced by $V(G)\backslash W$ is Hamilton-biconnected. In this paper, we prove that "Let $p\geq0$ and…