arXiv subjects
Publications and source records attributed to Weihua Yang.
Let $G$ be a contraction critically quasi $5$-connected graph on at least $14$ vertices. If there is a vertex $x\in V_{4}(G)$ such that $G[N_{G}(x)]\cong K_{1,3}$ or $G[N_{G}(x)]\cong C_{4}$, then $G$ has a quasi $5$-contractible subgraph $H$ such that $0<\|V(H)\|<4$.
A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil $ and gave some upper bounds on the minimum degree of the minimally $ t $-tough graphs in \cite{Katona, Gyula}. In this paper, we show that a minimally 1-tough graph $ G $ with girth $ g\geq 5 $ has minimum degree at most $ \lfloor\frac{n}{g+1}\rfloor+g-1$, and a minimally $ 1 $-tough graph with girth $ 4 $ has minimum degree at most $ \frac{n+6}{4}$. We also prove that the minimum degree of minimally $\frac{3}2$-tough claw-free graphs is $ 3 $.
Let $V_{8}+e$ denote the unique graph obtained from the Wagner graph, also known as $V_{8}$, by adding an edge between two vertices of distance 3 on the Hamilton cycle, which is exactly a split of a minor of the Petersen graph. A complete characterization of all internally 4-connected graphs with no $V_{8}$ minor is given in [J. Maharry and N. Robertson, The structure of graphs not topologically containing the Wagner graph, J. Combin. Theory Ser. B 121 (2016) 398-420]. In this paper we characterize all internally 4-connected graphs with no $V_{8}+e$ minor.
Let $F$ be an $(r+1)$-color critical graph with $r\geq 2$, that is, $χ(F)=r+1$ and there is an edge $e$ in $F$ such that $χ(F-e)=r$. Gerbner recently conjectured that every $n$-vertex maximal $F$-free graph with at least $(1-\frac{1}{r})\frac{n^2}{2}- o(n^{\frac{r+1}{r}})$ edges contains an induced complete $r$-partite graph on $n-o(n)$ vertices. Let $F_{s,k}$ be a graph obtained from $s$ copies of $C_{2k+1}$ by sharing a common edge. In this paper, we show that for all $k\geq 2$ if $G$ is an $n$-vertex maximal $F_{s,k}$-free graph with at least $n^{2}/4 - o(n^{\frac{s+2}{s+1}})$ edges, then $G$ contains an induced complete bipartite graph on $n-o(n)$ vertices. We also show that it is best possible. This disproves Gerbner's conjecture for $r=2$.
The small cycle decompositions of line graph ($λ$-fold line graph) of complete graphs and complete bipartite graphs are studied by many papers. In particular, Colby and Rodger obtained necessary and sufficient conditions for the existence of a $C_{4}$-decomposition of the $λ$-fold line graph of $K_{n}$, and Ganesamurthy and Paulraja completely determined the values of $n$ and $λ$ for which the $λ$-fold line graph of $K_{n}$ has a $C_{5}$-decomposition. In this paper, we obtain the necessary and sufficient condition for the existence of a $C_{4}$-decomposition of the $λ$-fold line graph of $K_{m,n}$.
A graph is near-bipartite if its vertex set can be partitioned into an independent set and a set which induces a forest. In this paper, planar graphs without cycles of length from 4 to 7 are shown to be near-bipartite.
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 $uv$ of $G$, $\{u,v\}$ is not a cut-set. Thus $G$ is 1-path Hamiltonian if $G\setminus \{u,v\}$ is connected for every edge $uv$ of $G$. Let $P=uvz$ be a 2-path of a 2-connected, $k$-regular graph $G$ on at most $2k$ vertices. In this paper, we show that there is a Hamiltonian cycle containing the 2-path $P$ if $G\setminus V(P)$ is connected. Therefore, the work implies a condition for a 2-connected, $k$-regular graph to be 2-path Hamiltonian. An example shows that the $2k$ is almost sharp, i.e., the number is at most $2k+1$.
Core decomposition is a fundamental operator in network analysis. In this paper, we study the problem of computing distance-generalized core decomposition on a network. A distance-generalized core, also termed $(k, h)$-core, is a maximal subgraph in which every vertex has at least $k$ other vertices at distance no larger than $h$. The state-of-the-art algorithm for solving this problem is based on a peeling technique which iteratively removes the vertex (denoted by $v$) from the graph that has the smallest $h$-degree. The $h$-degree of a vertex $v$ denotes the number of other vertices that are reachable from $v$ within $h$ hops. Such a peeling algorithm, however, needs to frequently recompute the $h$-degrees of $v$'s neighbors after deleting $v$, which is typically very costly for a large $h$. To overcome this limitation, we propose an efficient peeling algorithm based on a novel $h$-degree updating technique. Instead of recomputing the $h$-degrees, our algorithm can dynamically maintain the $h$-degrees for all vertices via exploring a very small subgraph, after peeling a vertex. We show that such an $h$-degree updating procedure can be efficiently implemented by an elegant bitmap technique. In addition, we also propose a sampling-based algorithm and a parallelization technique to further improve the efficiency. Finally, we conduct extensive experiments on 12 real-world graphs to evaluate our algorithms. The results show that, when $h\ge 3$, our exact and sampling-based algorithms can achieve up to $10\times$ and $100\times$ speedup over the state-of-the-art algorithm, respectively.
For any positive integer $k$, we show that every maximal $C_{2k+1}$-free graph with at least $n^2/4-o(n^{3/2})$ edges contains an induced complete bipartite subgraph on $(1-o(1))n$ vertices. We also show that this is best possible.
Let $Oct_{1}^{+}$ and $Oct_{2}^{+}$ be the planar and non-planar graphs that obtained from the Octahedron by 3-splitting a vertex respectively. For $Oct_{1}^{+}$, we prove that a 4-connected graph is $Oct_{1}^{+}$-free if and only if it is $C_{6}^{2}$, $C_{2k+1}^{2}$ $(k \geq 2)$ or it is obtained from $C_{5}^{2}$ by repeatedly 4-splitting vertices. We also show that a planar graph is $Oct_{1}^{+}$-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from $\{K_{1}, K_{2} ,K_{3}\} \cup \mathscr{K} \cup \{Oct,L_{5} \}$, where $\mathscr{K}$ is the set of graphs obtained by repeatedly taking the special 3-sums of $K_{4}$. For $Oct_{2}^{+}$, we prove that a 4-connected graph is $Oct_{2}^{+}$-free if and only if it is planar, $C_{2k+1}^{2}$ $(k \geq 2)$, $L(K_{3,3})$ or it is obtained from $C_{5}^{2}$ by repeatedly 4-splitting vertices.
Xiong and Liu [L. Xiong and Z. Liu, Hamiltonian iterated line graphs, Discrete Math. 256 (2002) 407-422] gave a characterization of the graphs $G$ for which the $n$-th iterated line graph $L^n(G)$ is hamiltonian, for $n\ge2$. In this paper, we study the existence of a hamiltonian path in $L^n(G)$, and give a characterization of $G$ for which $L^n(G)$ has a hamiltonian path. As applications, we use this characterization to give several upper bounds on the hamiltonian path index of a graph.
Zhu et al. [Theoret. Comput. Sci. 758 (2019) 1--8] introduced the $h$-edge tolerable diagnosability to measure the fault diagnosis capability of a multiprocessor system with faulty links. This kind of diagnosability is a generalization of the concept of traditional diagnosability. A graph is called a maximally connected graph if its minimum degree equals its vertex connectivity. It is well-known that many irregular networks are maximally connected graphs and the $h$-edge tolerable diagnosabilities of these networks are unknown, which is our motivation for research. In this paper, we obtain the lower bound of the $h$-edge tolerable diagnosability of a $t$-connected graph and establish the $h$-edge tolerable diagnosability of a maximally connected graph under the PMC model and the MM$^*$ model, which extends some results in [IEEE Trans. Comput. 23 (1974) 86--88], [IEEE Trans. Comput. 53 (2004) 1582--1590] and [Theoret. Comput. Sci. 796 (2019) 147--153].
We present a systematic calculation of the charged current semi-inclusive deeply inelastic scattering process at leading order twist-3 level in the parton model. We consider the general form of the negatively charged beam scattering off the polarized target which has spin-1. The calculations are carried out by applying the collinear expansion where multiple gluon scattering is taken into account and gauge links are obtained automatically. We first present the general form of the differential cross section in terms of the structure functions by kinematic analysis and then present the structure functions in terms of the the gauge invariant parton distribution functions up to twist-3 level. Considering the angle modulations and polarizations of the cross section, we calculate the complete azimuthal asymmetries in the charged current semi-inclusive deeply inelastic scattering process. The charge asymmetries are also considered in this paper with the introduction of the definitions of the $plus$ and $minus$ cross sections.
The $g$-extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464--473] determined the $3$-extra edge-connectivity of balanced hypercubes $BH_n$ and conjectured that the $g$-extra edge-connectivity of $BH_n$ is $λ_g(BH_n)=2(g+1)n-4g+4$ for $2\leq g\leq 2n-1$. In this paper, we confirm their conjecture for $n\geq 6-\dfrac{12}{g+1}$ and $2\leq g\leq 8$, and disprove their conjecture for $n\geq \dfrac{3e_g(BH_n)}{g+1}$ and $9\leq g\leq 2n-1$, where $e_g(BH_n)=\max\{|E(BH_n[U])|\mid U\subseteq V(BH_n), |U|=g+1\}$.
We present the complete azimuthal asymmetries at leading twist in terms of fragmentation functions in dihadron production semi-inclusive electron positron annihilation process. When the non-trivial $θ$ vacuum is taken into consideration, the parity symmetry of quantum chromodynamics is violated. As a consequence of the $local$ $\mathcal{P}$-odd effects, $\mathcal{P}$-odd fragmentation functions would contribute to the azimuthal asymmetries. Azimuthal asymmetry coming from two interference terms with opposite signs vanishes when sum over many events. This symmetry only survives on the event-by-event basis. Azimuthal asymmetry coming from two interference terms with same signs survives and can be measured to extract the $\mathcal{P}$-odd fragmentation functions. We also present the hadron polarizations.
Quantum chromodynamics is a fundamental non-abelian gauge theory of strong interactions. The physical quantum chromodynamics vacuum state, $|θ\rangle$, 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.
Quantum chromodynamics is a non-Abelian gauge theory of strong interactions, in which the parity symmetry can be violated by the non-trivial $θ$-vacuum tunneling effects. The $θ$-vacuum induces the local parity odd domains. Those reactions that occur in these domains can be affected by the tunneling effects and quantities become parity odd. In this paper we consider the fragmentation process where parity odd fragmentation functions are introduced. We present the fragmentation functions by decomposing the quark-quark correlator. Among the total 16 fragmentation functions, 8 of them are parity conserved, and the others are parity violated. They have a one-to-one correspondence. Positivity bounds of these one dimensional fragmentation functions are shown. To be explicit, we also introduce a operator definition of the parity odd correlator. According to the definition, we give a proof that the parity odd fragmentation functions are local quantities and vanish when sum over all the hadrons h.
Let $G$ be a graph on $n$ vertices. A linear forest is a graph consisting of vertex-disjoint paths and isolated vertices. A maximum linear forest of $G$ is a subgraph of $G$ with maximum number of edges, which is a linear forest. We denote by $l(G)$ this maximum number. Let $t=\left\lfloor (k-1)/2\right \rfloor$. Recently, Ning and Wang \cite{boning} proved that if $l(G)=k-1$, then for any $k<n$ \[ e(G) \leq \max \left\{\binom{k}{2},\binom{t}{2}+t (n - t)+ c \right\}, \] where $c=0$ if $k$ is odd and $c=1$ otherwise, and the inequality is tight. In this paper, we prove that if $l(G)=k-1$ and $δ(G)=δ$ ($δ<\lfloor k/2 \rfloor$), then for any $k<n$ \[ e(G) \leq \max \left\{\binom{k-δ}{2}+δ(n-k+δ),\binom{t}{2}+t\left(n-t\right)+c \right\}. \] When $δ=0$, it reduces to Ning and Wang's result. Moreover, let $r_3(G)$ be the number of triangles in $G$. We prove that if $l(G)=k-1$ and $δ(G)= δ$, then for any $k<n$ \[ r_3(G)\leq \max \left\{\binom{k-δ}{3}+\binomδ{2}(n-k+δ),\binom{t}{3}+\binom{t}{2}\left(n-t\right)+d \right\}. \] where $d=0$ if $k$ is odd and $d=t$ otherwise.