A kind of conditional connectivity of transposition networks generated by kk-trees

For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an $R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)- F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$, denoted by $\kappa^k(G)$, is the cardinality of the minimum $R_k$-vertex-cut of $G$, which is a refined measure for the fault tolerance of network $G$. In this paper, we study $\kappa^2$ for Cayley graphs generated by $k$-trees. Let $Sym(n)$ be the symmetric group on $\{1, 2, \cdots ,n\}$ and $\mathcal{T}$ be a set of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on $n$ vertices $\{1, 2, . . . ,n\}$ such that there is an edge $ij$ in $G(\mathcal{T})$ if and only if the transposition $ij\in \mathcal{T}$. The graph $G(\mathcal{T})$ is called the transposition generating graph of $\mathcal{T}$. We denote by $Cay(Sym(n),\mathcal{T})$ the Cayley graph generated by $G(\mathcal{T})$. The Cayley graph $Cay(Sym(n),\mathcal{T})$ is denoted by $T_kG_n$ if $G(\mathcal{T})$ is a $k$-tree. We determine $\kappa^2(T_kG_n)$ in this work. The trees are $1$-trees, and the complete graph on $n$ vertices is a $n-1$-tree. Thus, in this sense, this work is a generalization of the such results on Cayley graphs generated by transposition generating trees and the complete-transposition graphs.Comment: 11pages,2figure

Parity-odd Parton Distribution Functions from θ\theta-Vacuum

Quantum chromodynamics is a fundamental non-abelian gauge theory of strong interactions. The physical quantum chromodynamics vacuum state, $|\theta\rangle$, is a linear superposition of the $n$-vacua states with different topological numbers. Because of the configuration of the gauge fields, the tunneling events can induce the local parity-odd domains. Those interactions that occur in these domains can be affected by these effects. Considering the hadron (nucleon) system, we introduce the parity-odd parton distribution functions in order to describe the parity-odd structures inside the hadron in this paper. We obtain 8 parity-odd parton distribution functions at leading twist for spin-1/2 hadrons and present properties of these parton distribution functions. By introducing the parity-odd quark-quark correlator, we find the parity-odd effects vanish from the macroscopic point of view. Since the parity-odd effects are confined in small domains, we consider the high energy semi-inclusive deeply inelastic scattering process to investigate these effects by calculating the single spin asymmetries.Comment: arXiv admin note: text overlap with arXiv:1906.0342

On the existence of specified cycles in bipartite tournaments

For two integers $n\geq 3$ and $2\leq p\leq n$, we denote $D(n,p)$ the digraph obtained from a directed $n$-cycle by changing the orientations of $p-1$ consecutive arcs. In this paper, we show that a family of $k$-regular $(k\geq 3)$ bipartite tournament $BT_{4k}$ contains $D(4k,p)$ for all $2\leq p\leq 4k$ unless $BT_{4k}$ is isomorphic to a digraph $D$ such that $(1,2,3,...,4k,1)$ is a Hamilton cycle and $(4m+i-1,i)\in A(D)$ and $(i,4m+i+1)\in A(D)$, where $1\leq m\leq k-1$.Comment: 15 page

Embedding 5-planar graphs in three pages

A \emph{book-embedding} of a graph $G$ is an embedding of vertices of $G$ along the spine of a book, and edges of $G$ on the pages so that no two edges on the same page intersect. the minimum number of pages in which a graph can be embedded is called the \emph{page number}. The book-embedding of graphs may be important in several technical applications, e.g., sorting with parallel stacks, fault-tolerant processor arrays design, and layout problems with application to very large scale integration (VLSI). Bernhart and Kainen firstly considered the book-embedding of the planar graph and conjectured that its page number can be made arbitrarily large [JCT, 1979, 320-331]. Heath [FOCS84] found that planar graphs admit a seven-page book embedding. Later, Yannakakis proved that four pages are necessary and sufficient for planar graphs in [STOC86]. Recently, Bekos et al. [STACS14] described an $O(n^{2})$ time algorithm of two-page book embedding for 4-planar graphs. In this paper, we embed 5-planar graphs into a book of three pages by an $O(n^{2})$ time algorithm

The Tur\'an problem for a family of tight linear forests

Let $\mathcal{F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erd\H{o}s Matching Conjecture shows that $$ex_r(n,M_{k+1}^{(r)})= \max\left\{\binom{rk+r-1}{r},\binom{n}{r}-\binom{n-k}{r}\right\},$$ where $M_{k+1}^{(r)}$ represents the $r$-graph consisting of $k+1$ disjoint edges. Motivated by this conjecture, we consider the Tur\'an problem for tight linear forests. A tight linear forest is an $r$-graph whose connected components are all tight paths or isolated vertices. Let $\mathcal{L}_{n,k}^{(r)}$ be the family of all tight linear forests of order $n$ with $k$ edges in $r$-graphs. In this paper, we prove that for sufficiently large $n$, $$ex_r(n;\mathcal{L}_{n,k}^{(r)})=\max\left\{\binom{k}{r}, \binom{n}{r}-\binom{n-\left\lfloor (k-1)/r\right \rfloor}{r}\right\}+d,$$ where $d=o(n^r)$ and if $r=3$ and $k=cn$ with $0<c<1$, if $r\geq 4$ and $k=cn$ with $0<c<1/2$. The proof is based on the weak regularity lemma for hypergraphs. We also conjecture that for arbitrary $k$ satisfying $k \equiv 1\ (mod\ r)$, the error term$d$ in the above result equals 0. We prove that the proposed conjecture implies the Erd\H{o}s Matching Conjecture directly

The Tur\'{a}n Number for Spanning Linear Forests

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on $n$ vertices, then we often call the problem a spanning Tur\'{a}n problem. A linear forest is a graph whose connected components are all paths and isolated vertices. In this paper, we let $\mathcal{L}_n^k$ be the set of all linear forests of order $n$ with at least $n-k+1$ edges. We prove that when $n\geq 3k$ and $k\geq 2$, $$ex(n;\mathcal{L}_n^k)=\binom{n-k+1}{2}+ O(k^2).$$ Clearly, the result is interesting when $k=o(n)$

Electronic Conduction in Short DNA Wires

A strict method is used to calculate the current-voltage characteristics of a double-stranded DNA. A more reliable model considering the electrostatic potential drop along an individual DNA molecular wire between the contacts is considered and the corresponding Green's Function is obtained analytically using Generating Function method, which avoids difficult numerical evaluations. The obtained results indicate that the electrostatic drop along the wire always increases the conductor beyond the threshold than without considering it, which is in agreement with recent experiments. The present method can also be used to calculate the current-voltage characteristics for other molecular wires of arbitrary length.Comment: 9 pages, 2 figur

A new transformation into State Transition Algorithm for finding the global minimum

To promote the global search ability of the original state transition algorithm, a new operator called axesion is suggested, which aims to search along the axes and strengthen single dimensional search. Several benchmark minimization problems are used to illustrate the advantages of the improved algorithm over other random search methods. The results of numerical experiments show that the new transformation can enhance the performance of the state transition algorithm and the new strategy is effective and reliable.Comment: 5 pages, 6 figure

On the spanning connectivity of tournaments

Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$ internally disjoint paths with the same direction (resp. with different directions allowed) between $u$ and $v$ and it contains all vertices of $D$. A digraph $D$ is $k^{*}$-strongly (resp. $k^{*}$-weakly) connected if there exists a strong (resp. weak) $k^{*}$-container between any two distinct vertices. We define the strong (resp. weak) spanning connectivity of a digraph $D$, $\kappa_{s}^{*}(D)$ (resp. $\kappa_{w}^{*}(D)$ ), to be the largest integer $k$ such that $D$ is $\omega^{*}$-strongly (resp. $\omega^{*}$-weakly) connected for all $1\leq \omega\leq k$ if $D$ is a $1^{*}$-strongly (resp. $1^{*}$-weakly) connected. In this paper, we show that a tournament with $n$ vertices and irregularity $i(T)\leq k$, if $n\geq6t+5k$ $(t\geq2)$, then $\kappa_{s}^{*}(T)\geq t$ and $\kappa_{w}^{*}(T)\geq t+1$ if $n\geq6t+5k-3$ $(t\geq2)$.Comment: 11 page

On the number of proper paths between vertices in edge-colored hypercubes

Given an integer $1\leq j <n$, define the $(j)$-coloring of a $n$-dimensional hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all edges in dimension $i$, $1\leq i \leq j$, have color $1$ and all other edges have color $2$. Cheng et al. [Proper distance in edge-colored hypercubes, Applied Mathematics and Computation 313 (2017) 384-391.] determined the number of distinct shortest properly colored paths between a pair of vertices for the $(1)$-colored hypercubes. It is natural to consider the number for $(j)$-coloring, $j\geq 2$. In this note, we determine the number of different shortest proper paths in $(j)$-colored hypercubes for arbitrary $j$.Comment: 9 page