Searcharxiv⌕ Search

arXiv subjects

Dong Ye

Publications and source records attributed to Dong Ye.

At least 55 records · Page 3Linked to original sources
arXiv2019

Connectivity for Kite-Linked Graphs

For a given graph $H$, a graph $G$ is $H$-linked if, for every injection $φ: V(H) \to V(G)$, the graph $G$ contains a subdivision of $H$ with $φ(v)$ corresponding to $v$, for each $v\in V(H)$. Let $f(H)$ be the minimum integer $k$ such that every $k$-connected graph is $H$-linked. Among graphs $H$ with at least four vertices, the exact value $f(H)$ is only know when $H$ is a path with four vertices or a cycle with four vertices. A kite is graph obtained from $K_4$ by deleting two adjacent edges, i.e., a triangle together with a pendant edge. Recently, Liu, Rolek and Yu proved that every $8$-connected graph is kite-linked. The exact value of $f(H)$ when $H$ is the kite remains open. In this paper, we settle this problem by showing that every 7-connected graph is kite-linked.

math.CO↗
arXiv2019

Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ Δ^m u = \pm u^α \quad \text{ in } \mathbb R^n \] with $n \geqslant 1$, $m \geqslant 1$, and $α\in \mathbb R$. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants $n$, $m$ and $α$. Remarkably, we are able to provide necessary and sufficient conditions on the exponent $α$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive.

math.AP↗
arXiv2019

Toughness and spanning trees in $K_4$-minor-free graphs

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the number of components of a graph $G$. We show that if $G$ is a connected $K_4$-minor-free graph and $$ c(G-S) \;\le\; \sum_{v \in S} (f(v)-1) \quad\hbox{for all $S \subseteq V(G)$ with $S \ne \emptyset$} $$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a $\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree. These results are stronger than results for general graphs due to Win (for $k$-trees) and Ellingham, Nam and Voss (for $f$-trees). The $K_4$-minor-free graphs form a subclass of planar graphs, and are identical to graphs of treewidth at most $2$, and also to graphs whose blocks are series-parallel. We provide examples to show that the inequality above cannot be relaxed by adding $1$ to the right-hand side, and also to show that our result does not hold for general planar graphs. Our proof uses a technique where we incorporate toughness-related information into weights associated with vertices and cutsets.

math.CO↗
arXiv2019

Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains

We consider blow-up results for a system of inhomogeneous wave inequalities in exterior domains. We will handle three type boundary conditions: Dirichlet type, Neumann type and mixed boundary conditions. We use a unified approach to show the optimal criteria of Fujita type for each case. Our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation. We provide many new results and close some open questions.

math.AP↗
arXiv2019

Cycle Traversability for Claw-free Graphs and Polyhedral Maps

Let $G$ be a graph, and $v\in V(G)$ and $S\subseteq V(G)\backslash v$ of size at least $k$. An important result on graph connectivity due to Perfect states that, if $v$ and $S$ are $k$-linked, then a $(k-1)$-link between a vertex $v$ and $S$ can be extended to a $k$-link between $v$ and $S$ such that the endvertices of the $(k-1)$-link are also the endvertices of the $k$-link. We begin by proving a generalization of Perfect's result by showing that, if two disjoint sets $S_1$ and $S_2$ are $k$-linked, then a $t$-link ($t< k$) between two disjoint sets $S_1$ and $S_2$ can be extended to a $k$-link between $S_1$ and $S_2$ such that the endvertices of the $t$-link are preserved in the $k$-link. Next, we are able to use these results to show that a 3-connected claw-free graph always has a cycle passing through any given five vertices but avoiding any other one specified vertex. We also show that this result is sharp by exhibiting an infinite family of 3-connected claw-free graphs in which there is no cycle containing a certain set of six vertices but avoiding a seventh specified vertex. A direct corollary of our main result shows that, a 3-connected claw-free graph has a topological wheel minor $W_k$ with $k\le 5$ if and only if it has a vertex of degree at least $k$. Finally, we also show that a graph polyhedrally embedded in a surface always has a cycle passing through any given three vertices but avoiding any other specified vertex. The result is best possible in the sense that the polyhedral embedding assumption is necessary, and there are infinitely many graphs polyhedrally embedded in any surface having no cycle containing a certain set of four vertices but avoiding a fifth specified vertex.

math.CO↗
arXiv2019

Dominating maximal outerplane graphs and Hamiltonian plane triangulations

Let $G$ be a graph and $γ(G)$ denote the domination number of $G$, i.e. the cardinality of a smallest set of vertices $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. Matheson and Tarjan conjectured that a plane triangulation with a sufficiently large number $n$ of vertices has $γ(G)\le n/4$. Their conjecture remains unsettled. In the present paper, we show that: (1) a maximal outerplane graph with $n$ vertices has $γ(G)\le \lceil \frac{n+k} 4\rceil$ where $k$ is the number of pairs of consecutive degree 2 vertices separated by distance at least 3 on the boundary of $G$; and (2) a Hamiltonian plane triangulation $G$ with $n \ge 23$ vertices has $γ(G)\le 5n/16 $. We also point out and provide counterexamples for several recent published results of Li et al in [Discrete Appl. Math.198 (2016) 164-169] on this topic which are incorrect.

math.CO↗
arXiv2019

On Gaussian curvature equations in $\mathbb{R}^2$ with prescribed non-positive curvature

The purpose of this paper is to study the solutions of $$ Δu +K(x) e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2 $$ with $K\le 0$. We introduce the following quantity: $$α_p(K)=\sup\left\{α\in \mathbb{R}:\, \int_{\mathbb{R}^2} |K(x)|^p(1+|x|)^{2αp+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1.$$ Under the assumption $({\mathbb H}_1)$: $α_p(K)> -\infty$ for some $p>1$ and $α_1(K) > 0$, we show that for any $0 < α< α_1(K)$, there is a unique solution $u_α$ with $u_α(x) = α\ln |x|+ c_α+o\big(|x|^{-\frac{2β}{1+2β}} \big)$ at infinity and $β\in (0,\,α_1(K)-α)$. Furthermore, we show an example $K_0 \leq 0$ such that $α_p(K_0) = -\infty$ for any $p>1$ and $α_1(K_0) > 0$, for which we study the asymptotic behavior of solutions. In particular, we prove the existence of a solution $u_*$ such that $u_* -α_*\ln|x| = O(1)$ at infinity for some $α_* > 0$, but who does not converge to a constant at infinity. This example exhibits a new phenomenon of solutions with logarithmic growth and non-uniform behavior at infinity.

math.AP↗
arXiv2018

Minimum $T$-Joins and Signed-Circuit Covering

Let $G$ be a graph and $T$ be a vertex subset of $G$ with even cardinality. A $T$-join of $G$ is a subset $J$ of edges such that a vertex of $G$ is incident with an odd number of edges in $J$ if and only if the vertex belongs to $T$. Minimum $T$-joins have many applications in combinatorial optimizations. In this paper, we show that a minimum $T$-join of a connected graph $G$ has at most $|E(G)|-\frac 1 2 |E(\widehat{\, G\,})|$ edges where $\widehat{\,G\,}$ is the maximum bidegeless subgraph of $G$. Further, we are able to use this result to show that every flow-admissible signed graph $(G,σ)$ has a signed-circuit cover with length at most $\frac{19} 6 |E(G)|$. Particularly, a 2-edge-connected signed graph $(G,σ)$ with even negativeness has a signed-circuit cover with length at most $\frac 8 3 |E(G)|$.

math.CO↗
arXiv2017

On the anti-Kelulé problem of cubic graphs

An edge set $S$ of a connected graph $G$ is called an anti-Kekulé set if $G-S$ is connected and has no perfect matchings, where $G-S$ denotes the subgraph obtained by deleting all edges in $S$ from $G$. The anti-Kekulé number of a graph $G$, denoted by $ak(G)$, is the cardinality of a smallest anti-Kekulé set of $G$. It is NP-complete to find the smallest anti-Kekulé set of a graph. In this paper, we show that the anti-Kekulé number of a 2-connected cubic graph is either 3 or 4, and the anti-Kekulé number of a connected cubic bipartite graph is always equal to 4. Furthermore, a polynomial time algorithm is given to find all smallest anti-Kekulé sets of a connected cubic graph.

math.CO↗
arXiv2017

Resonance Graphs and Perfect Matchings of Graphs on Surfaces

Let $G$ be a graph embedded in a surface and let $\mathcal F$ be a set of even faces of $G$ (faces bounded by a cycle of even length). The resonance graph of $G$ with respect to $\mathcal F$, denoted by $R(G;\mathcal F)$, is a graph such that its vertex set is the set of all perfect matchings of $G$ and two vertices $M_1$ and $M_2$ are adjacent to each other if and only if the symmetric difference $M_1\oplus M_2$ is a cycle bounding some face in $\mathcal F$. It has been shown that if $G$ is a matching-covered plane bipartite graph, the resonance graph of $G$ with respect to the set of all inner faces is isomorphic to the covering graph of a distributive lattice. It is evident that the resonance graph of a plane graph $G$ with respect to an even-face set $\mathcal F$ may not be the covering graph of a distributive lattice. In this paper, we show the resonance graph of a graph $G$ on a surface with respect to a given even-face set $\mathcal F$ can always be embedded into a hypercube as an induced subgraph. Furthermore, we show that the Clar covering polynomial of $G$ with respect to $\mathcal F$ is equal to the cube polynomial of the resonance graph $R(G;\mathcal F)$, which generalizes previous results on some subfamilies of plane graphs.

math.CO↗
arXiv2017

On Perfect Matchings in Matching Covered Graphs

Let $G$ be a matching-covered graph, i.e., every edge is contained in a perfect matching. An edge subset $X$ of $G$ is feasible if there exists two perfect matchings $M_1$ and $M_2$ such that $|M_1\cap X|\not\equiv |M_2\cap X| \pmod 2$. Lukot'ka and Rollová proved that an edge subset $X$ of a regular bipartite graph is not feasible if and only if $X$ is switching-equivalent to $\emptyset$, and they further ask whether a non-feasible set of a regular graph of class 1 is always switching-equivalent to either $\emptyset$ or $E(G)$? Two edges of $G$ are equivalent to each other if a perfect matching $M$ of $G$ either contains both of them or contains none of them. An equivalent class of $G$ is an edge subset $K$ with at least two edges such that the edges of $K$ are mutually equivalent. An equivalent class is not a feasible set. Lovász proved that an equivalent class of a brick has size 2. In this paper, we show that, for every integer $k\ge 3$, there exist infinitely many $k$-regular graphs of class 1 with an arbitrarily large equivalent class $K$ such that $K$ is not switching-equivalent to either $\emptyset$ or $E(G)$, which provides a negative answer to the problem proposed by Lukot'ka and Rollová. Further, we characterize bipartite graphs with equivalent class, and characterize matching-covered bipartite graphs of which every edge is removable.

math.CO↗
arXiv2017

Using single nanoparticle tracking obtained by nanophotonic force microscopy to simultaneously characterize nanoparticle size distribution and nanoparticle-surface interactions

Comprehensive characterization of nanomaterials for medical applications is a challenging and complex task due to the multitude of parameters which need to be taken into consideration in a broad range of conditions. Routine methods such as dynamic light scattering or nanoparticle tracking analysis provide some insight into the physicochemical properties of particle dispersions. For nanomedicine applications the information they supply can be of limited use. For this reason, there is a need for new methodologies and instruments that can provide additional data on nanoparticle properties such as their interactions with surfaces. Nanophotonic force microscopy has been shown as a viable method for measuring the force between surfaces and individual particles in the nano-size range. Here we outline a further application of this technique to measure the size of single particles and based on these measurement build the distribution of a sample. We demonstrate its efficacy by comparing the size distribution obtained with nanophotonic force microscopy to established instruments, such as dynamic light scattering and differential centrifugal sedimentation. Our results were in good agreement to those observed with all other instruments. Furthermore, we demonstrate that the methodology developed in this work can be used to study complex particle mixtures and the surface alteration of materials. For all cases studied, we were able to obtain both the size and the interaction potential of the particles with a surface in a single measurement.

cond-mat.soft↗
arXiv2017

Connectivity and $W_v$-Paths in Polyhedral Maps on Surfaces

The $W_v$-Path Conjecture due to Klee and Wolfe states that any two vertices of a simple polytope can be joined by a path that does not revisit any facet. This is equivalent to the well-known Hirsch Conjecture. Klee proved that the $W_v$-Path Conjecture is true for all 3-polytopes (3-connected plane graphs), and conjectured even more, namely that the $W_v$-Path Conjecture is true for all general cell complexes. This general $W_v$-Path Conjecture was verified for polyhedral maps on the projective plane and the torus by Barnette, and on the Klein bottle by Pulapaka and Vince. Let $G$ be a graph polyhedrally embedded in a surface $Σ$, and $x, y$ be two vertices of $G$. In this paper, we show that if there are three internally disjoint $(x,y)$-paths which are homotopic to each other, then there exists a $W_v$-path joining $x$ and $y$. For every surface $Σ$, define a function $f(Σ)$ such that if for every graph polyhedrally embedded in $Σ$ and for a pair of vertices $x$ and $y$ in $V(G)$, the local connectivity $κ_G(x,y) \ge f(Σ)$, then there exists a $W_v$-path joining $x$ and $y$. We show that $f(Σ)=3$ if $Σ$ is the sphere, and for all other surfaces $3-τ(Σ)\le f(Σ)\le 9-4χ(Σ)$, where $χ(Σ)$ is the Euler characteristic of $Σ$, and $τ(Σ)=χ(Σ)$ if $χ(Σ)< -1$ and 0 otherwise. Further, if $x$ and $y$ are not cofacial, we prove that $G$ has at least $κ_G(x,y)+4χ(Σ)-8$ internally disjoint $W_v$-paths joining $x$ and $y$. This bound is sharp for the sphere. Our results indicate that the $W_v$-path problem is related to both the local connectivity $κ_G(x,y)$, and the number of different homotopy classes of internally disjoint $(x,y)$-paths as well as the number of internally disjoint $(x,y)$-paths in each homotopy class.

math.CO↗
arXiv2016

Inverses of Bipartite Graphs

Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39].

math.CO↗
arXiv2016

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan, and Vardanyan that every graph $G$ with $Δ(G)-δ(G) \le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. In this note, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.

math.CO↗
arXiv2016

Circuit Covers of Cubic Signed Graphs

A signed graph is a graph $G$ associated with a mapping $σ: E(G)\to \{-1,+1\}$, denoted by $(G,σ)$. A $cycle$ of $(G,σ)$ is a connected 2-regular subgraph. A cycle $C$ is $positive$ if it has an even number of negative edges, and negative otherwise. A $circuit$ of of a signed graph $(G,σ)$ is a positive cycle or a barbell consisting of two edge-disjoint negative cycles joined by a path. The definition of a circuit of signed graph comes from the signed-graphic matroid. A circuit cover of $(G,σ)$ is a family of circuits covering all edges of $(G,σ)$. A circuit cover with the smallest total length is called a shortest circuit cover of $(G,σ)$ and its length is denoted by $\text{scc}(G,σ)$. Bouchet proved that a signed graph with a circuit cover if and only if it is flow-admissible (i.e., has a nowhere-zero integer flow). Máčajová et. al. show that a 2-edge-connected signed graph $(G,σ)$ has $\text{scc}(G,σ)\le 9 |E(G)|$ if it is flow-admissible. This bound was improved recently by Cheng et. al. to $\text{scc}(G,σ) \le 11|E(G)|/3$ for 2-edge-connected signed graphs with even negativeness, and particularly, $\text{scc}(G,σ)\le 3|E(G)|+ε(G,σ)/3$ for 2-edge-connected cubic signed graphs with even negativeness (where $ε(G,σ)$ is the negativeness of $(G,σ)$). In this paper, we show that every 2-edge-connected cubic signed graph has $\text{scc}(G,σ)\le 26|E(G)|/9$ if it is flow-admissible, and $\text{scc}(G,σ)\le 23|E(G)|/9$ if it has even negativeness.

math.CO↗