arXiv subjects
Publications and source records attributed to Jin Yan.
Let $1 0$, and $p\in\mathbb R$. We study positive $C^1_{\mathrm{loc}}$ weak solutions of \[-Δ_m u=u^p|Du|^q\qquad\text{in }\mathbb R^n.\] If $0 m$ follows from the theorem of Lu and Zhu.
Oriented graph discrepancy problems focus on finding specific subgraphs within a given oriented graph $G$ that contain a significant number of edges in one direction. Gishboliner, Krivelevich, and Michaeli initiated the study of oriented discrepancy for Hamilton cycles, and Freschi and Lo subsequently proved a sharp minimum-degree theorem and asked for an Ore-type analogue. For an oriented graph $G$ on $n$ vertices, define $σ_2(G):=\min\{d_G(x)+d_G(y):x\neq y,\ xy,yx\notin E(G)\},$ with $σ_2(G):=2(n-1)$ when $G$ is a tournament. Given a cycle $C$ in an oriented graph, let $σ_{\max}(C)$ denote the maximum number of edges oriented consistently with one of the two traversal directions of $C$. We prove that for every $γ>0$ there exists $n_0$ such that every oriented graph $G$ on $n\geq n_0$ vertices with $σ_2(G)\geq n$ contains a Hamilton cycle $C$ satisfying $$ σ_{\max}(C)\geq\max\left\{\frac n2,\frac{σ_2(G)}2-γn\right\}. $$ This partially answers a question of Freschi and Lo and gives an asymptotic affirmative answer to a conjecture of Ai et al. The bound is asymptotically best possible.
Garamvölgyi, Jordán, Király and Villányi [{{\bf Forum Math. Pi} \textbf{13} (2025), Paper No.~e11}] posed two sharp connectivity conjectures for packing rigid spanning subgraphs: one for the equal-dimensional case and the other for the packing of a $d$-rigid spanning subgraph with a spanning tree. We prove a unified theorem: for arbitrary positive integers $d_1,\ldots,d_s$, every $\sum_{i=1}^{s}d_i(d_i+1)$-connected graph contains pairwise edge-disjoint spanning subgraphs $H_1,\ldots,H_s$ such that $H_i$ is $d_i$-rigid for every $i$. The connectivity bound is sharp whenever $\sum_{i=1}^{s}d_i(d_i+1)\ge4$. As special cases, the theorem settles both conjectures, confirms the conjecture of Garamvölgyi, Jordán and Király [{\bf J. Combin. Theory Ser. B} \textbf{166} (2024), 1--29] that every $tk(k+1)$-connected graph contains $t$ pairwise edge-disjoint $k$-connected spanning subgraphs, and gives the sharp threshold $d(d+1)+2r$ for packing one $d$-rigid spanning subgraph together with $r$ pairwise edge-disjoint spanning trees. We also obtain two upper bounds related to Thomassen's conjecture on highly connected orientations of graphs. If $f(q)$ is the least integer such that every $f(q)$-connected graph has a $q$-connected orientation, then $f(q)\le(25q^2+41q-16)/2$ for every $q\ge3$ and $f(q)\le8q^2+212q+1404=(8+o(1))q^2$ for all sufficiently large $q$; these two results reduce the leading coefficient in the previous quadratic bound from $320$ to $25/2$ for every $q\ge3$ and $8$ for all sufficiently large $q$. Compared to the bound for $f(q)$ obtained by Garamvölgyi et al. we obtain the better bounds, not only through our tight rigidity result but also by exploiting the leftover edges when we remove two edge-disjoint spanning (sufficiently) rigid graphs.
We prove interior Hessian estimates in arbitrary dimensions for admissible solutions of Hessian quotient equations with constant right-hand side, when the numerator and denominator differ in degree by one or two. Counterexamples show that this degree range is optimal. The proof combines a new nonlinear comparison function, a doubling maximum principle, and a Pogorelov estimate.
We resolve the long-standing problem of establishing interior \(C^2\) estimates for admissible solutions of the graphical scalar curvature equation in every dimension \(n\ge 3\). More precisely, we prove interior curvature estimates for admissible solutions to the constant graphical scalar curvature equation. The proof combines Jacobi inequalities with a two-surface maximum principle and a two-surface Pogorelov estimate.
Let $r\ge2$, let $T_r$ denote the transitive tournament on $r$ vertices, and write $d_G^*(v):=\max\{d_G^+(v),d_G^-(v)\}$. We prove that if $r\mid n$ and an $n$-vertex digraph $G$ satisfies $d_G^*(x)+d_G^*(y)\ge 2(1-1/r)n-1$ for every $x\ne y \in V(G)$ with $xy \notin E(G)$, then $G$ has a $T_r$-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree $d^*_G(x) \ge (1-1/r)n$ and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n - 1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollobás--Erdős--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2α(G)+δ(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $κ(G)\geρn$ and $ρ>1/3$ has a hitting set of size at most $\lfloor(ρ-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $κ(G)>4n/9$ and to the sharp bound $3$ when $κ(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=Ω(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.
We study two open path problems with prescribed endpoints posed by Bang-Jensen and Gutin. The first asks for a longest $(x,y)$-path in a semicomplete digraph. The second asks whether a locally semicomplete digraph has a Hamiltonian $(x,y)$-path. For semicomplete digraphs, we solve the first problem when the endpoints lie in different strong components. We also prove that if a non-Hamiltonian longest $(x,y)$-path omits a set of vertices, then these vertices together with $x$ and $y$ have a Hamiltonian $(y,x)$-path. This gives an equivalent cycle problem. We then give an exact algorithm that runs in polynomial time when the number of omitted vertices is fixed. For locally semicomplete digraphs, we determine the possible endpoints in the connected nonstrong case. Known results then leave only strong, nonsemicomplete, non-$4$-strong digraphs unresolved. Every such digraph of order at least five has a strong vertex cut of size at most three. Two examples show that a spanning directed path together with a vertex-disjoint directed cycle is not sufficient, and that vertices of one strong component need not occur consecutively on a Hamiltonian path.
In this paper, we study the asymptotic Plateau problem in hyperbolic space for hypersurfaces of constant $H_k$-curvature. We prove the existence of a smooth complete $k$-convex hypersurface in $\mathbb{H}^{n+1}$ satisfying \[ H_k(κ)=σ, \qquad σ\in(0,1), \] with prescribed asymptotic boundary at infinity. In particular, our result extends the range of the constant $σ$ in the existence theorem of Guan and Spruck [J. Eur. Math. Soc. 12 (2010), no. 3, 797--817] for $H_{k}$ curvature to the full interval $(0,1)$.
A digraph is \textbf{supereulerian} if it contains a spanning closed trail. Let $α_2(D)$ denote the maximum cardinality of a vertex set inducing no 2-cycle. In this paper, we characterize supereulerianity in a strong digraph $D$ with $α_2(D)=4$ by proving that a strong digraph $D$ with $α_2(D)=4$ and $λ(D)\ge 2$ is supereulerian if and only if $D$ does not belong to an exceptional family $\mathcal H$ of $2$-arc-strong digraphs with $α_2(D)=4$. Furthermore, every strong digraph satisfying $α_2(D)=4$ and $λ(D)\geq3$ is supereulerian.
Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [\emph{SIAM Journal on Discrete Mathematics} \textbf{40} (2026), 988--999]. An incidence-based reduction to their bidirected packing--covering theorem preserves the packing value and projects transversals without increasing their cardinality.
We establish a new concavity inequality for the elementary symmetric function \(σ_k\), which controls the spectral quadratic form arising from the second variation of \(\logλ_{\max}\). The proof is based on an easy--hard decomposition of the spectral variables. The easy region is treated using an optimal constrained concavity estimate for \(σ_k\), whereas the hard region is analyzed through Gårding-root coordinates and the concavity of the ordered partial sums of the inverse roots. As an application, for \(n/2\leq k<n\), we obtain global curvature estimates for closed star-shaped \(k\)-convex hypersurfaces satisfying \(σ_k(κ)=f(X,ν)\) with a general positive right-hand side, together with the corresponding global-to-boundary estimates for Euclidean Hessian equations.
For two digraph properties $P_1$ and $P_2$, a $(P_1,P_2)$-arc-decomposition of a digraph $D$ is a partition $A(D)=A_1\mathbin{\dot\cup}A_2$ such that the spanning subdigraphs $D[A_1]$ and $D[A_2]$ have properties $P_1$ and $P_2$, respectively. For example, a (strong,strong)-arc-decomposition of a digraph $D=(V,A)$ is a partitioning $A=A_1\cup{}A_2$ of $A$ so that each of the spanning digraphs $D_i=(V,A_i)$, $i=1,2$ are strongly connected. We prove that it is NP-complete to decide whether a digraph admits an arc-decomposition with properties $(P_1,P_2)$ where $(P_1,P_2)\in \{$(is a perfect matching, having no odd directed cycle), (perfect matching, strong), (perfect matching, having an out-branching), (is a cycle factor, having no odd directed cycle)$\}$. These results settle some open problems posed by Bang-Jensen, Bessy, Gonçalves, and Picasarri-Arrieta [Theoret. Comput. Sci. 928 (2022), 167--182].
The low-lying isomeric state of $^{229m}$Th, owing to its unique nuclear energy structure, has been widely regarded as one of the most promising candidates for the development of a nuclear clock. However, the limited availability of suitable $^{229}$Th sources with sufficiently high activity remains a major challenge for experimental investigations of the $^{229m}$Th isomer. We propose a neutron-capture-based approach for the in-situ production of $^{229m,g}$Th by doping $^{228}$Ra into crystal hosts, where $^{229m,g}$Th is generated through neutron-capture reactions followed by a sequence of radioactive decay processes. We systematically investigate the background contributions associated with the three doped crystal hosts, namely CaF$_2$, SrF$_2$, and LiF, and evaluate their impact on the detection and identification of $^{229m}$Th. Under a neutron flux of $10^{15}\ \mathrm{n/cm^{2}/s}$ and a $^{228}$Ra doping concentration of $10^{19}\ \mathrm{cm^{-3}}$, the proposed method is capable of producing on the order of $10^{12}$ $^{229}$Th and $^{229m}$Th nuclei within only 1 s of irradiation, with a signal-to-noise ratios as high as $10^5$. In addition, the influences of detector wavelength resolution and post-irradiation measurement time on the detectability of the $^{229m}$Th signal are systematically analyzed, and the corresponding optimal measurement conditions are identified. Furthermore, the spatial distribution of neutron-produced $^{229}$Th within the crystal is investigated, providing practical guidance for optimizing crystal geometry and illumination configuration in future continuous-wave VUV absorption spectroscopy experiments. These results suggest that the proposed scheme provides a promising alternative pathway for the production and detection of $^{229\mathrm{m,g}}$Th, which may facilitate future studies toward the realization of nuclear-clock-based technologies.
We study spanning $H$-subdivisions in dense graphs where the length of every subdivision path is prescribed in advance. This problem is motivated in part by a question of Pavez-Signé [Combin. Probab. Comput. 33 (2024), 121--128], who asked whether the subdivision paths in a spanning $H$-subdivision can be required to have similar lengths. Let $h\ge3$ be an integer and let $0<β\llα\ll1/h$. We prove that, for all sufficiently large $n$, every $n$-vertex graph $G$ with $δ(G)\ge n/2+\lfloor h/3\rfloor$ has the following property. For every graph $H$ with $h$ edges and no isolated vertices, write $E(H)=\{e_1,\ldots,e_h\}$, and every choice of integers $\ell_1,\ldots,\ell_h\ge4$ satisfying $\sum_{i=1}^h\ell_i=n-|V(H)|+h$ and $\sum_{\ell_i<αn}\ell_i\leβn$, the graph $G$ contains a spanning $H$-subdivision in which the $i$th edge of $H$ is replaced by a path of length exactly $\ell_i$. We also give a family of examples showing that a linear additive term in $h$ is necessary in general.
Pavez-Signé [Combin. Probab. Comput. 33 (2024), 121--128] conjectured a Dirac-type condition for spanning $H$-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths. Lee [European J. Combin. 124 (2025), 104059] resolved the existence conjecture in the stronger setting of digraphs. We answer the length-control question in this stronger directed setting: for every $\varepsilon>0$, there exists a constant $C_0>0$ such that, for every digraph $H$ with $h$ arcs and no isolated vertices, every $n$-vertex digraph $D$ with $n\ge C_0h$ and $δ^0(D)\ge(1/2+\varepsilon)n$ contains a spanning $H$-subdivision whose subdivision paths have lengths differing by at most one.
Jackson and Thomassen conjectured that every $2k$-strong digraph admits a spanning $k$-strong oriented subdigraph [Ann. N. Y. Acad. Sci. 555 (1989) 402-412]. The conjecture holds for $k=1$ but other than some partial results that have been obtained for general $k$ in some special families of digraphs, including symmetric digraphs, the conjecture remains wide open in general. Even the existence of an integer $K$ such that every $K$-strong digraph has a 2-strong spanning oriented subdigraph is open. As a natural optimization counterpart, the minimum spanning $k$-strong subdigraph (MSSS$_k$) problem, is to find the minimum number of arcs in a spanning $k$-strong subdigraph of a $k$-strong digraph. This problem is NP-hard already for $k=1$ as it generalizes the hamiltonian cycle problem. In this paper, we address both problems simultaneously for the class of split digraphs, by constructing sparse spanning $k$-strong oriented subdigraphs. Specifically, we prove that every $k$-strong split digraph $D = (V_1, V_2; A)$ with minimum semi-degree {$δ^0(D)\geq 26k+15$} contains a spanning $k$-strong oriented subdigraph with no more than $kn+k|V_1|+98k^2+38k+3$ arcs, where $kn + k|V_1|$ is tight and the $k^{2}$ term is tight up to a constant factor. For the class of $k$-strong tournaments with minimum semi-degree at least $26k+15$ our results improve the bound obtained by Kang in [Combin. Probab. Comput., 27:892-907, 2018].
Let $k$ be a positive integer. Jackson and Thomassen conjectured in 1989 that there exists an integer function $f(k)$ such that every $f(k)$-strong digraph admits a spanning $k$-strong oriented subdigraph. They even conjectured that one can take $f(k)=2k$ [Ann. N. Y. Acad. Sci. 555 (1989) 402-412]. Already the existence of $f(2)$ is open for general digraphs. Thomassen proved that $f(2)=4$ for symmetric digraphs. For general $k$, the existence of $f(k)$ was only known for locally semicomplete digraphs and quasi-transitive digraphs. Guo proved that every ${(3k-2)}$-strong locally semicomplete digraph contains a spanning $k$-strong local tournament [Discrete Appl. Math. 79 (1997) 119--125]. One can deduce from Guo's result that we have $f(k)\leq 3k-2$ for quasi-transitive digraphs. In this paper, we prove the existence of $f(k)$ for two subclasses of the semicomplete multipartite digraphs, namely extended semicomplete digraphs and semicomplete split digraphs. We prove that every $(4k+1)$-strong extended semicomplete digraph contains a spanning $k$-strong oriented subdigraph and every $5k$-strong semicomplete split digraph contains a spanning $k$-strong oriented subdigraph. The first result implies that for the large class of digraphs which can be obtained from some semicomplete digraph $S$ on at least 3 vertices by substituting arbitrary digraphs for each vertex of $S$ we also have $f(k)\leq 4k+1$.