arXiv subjects

Qinghong Zhao

Publications and source records attributed to Qinghong Zhao.

9 recordsLinked to original sources

arXiv2026

Multicolor Ramsey and list Ramsey numbers for star-like trees

For a graph $H$, the $k$-color Ramsey number $r(H;k)$ is the least integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $H$. A $k$-list assignment has $|L(e)|=k$ for every edge. The list Ramsey number $r_\ell(H;k)$ is the least integer $N$ for which there exists a $k$-list assignment on $E(K_N)$ such that every coloring from the lists contains a monochromatic copy of $H$. Let $K_{1,n}$ be a star, $S(n,m)$ the double star obtained by joining the centers of $K_{1,n}$ and $K_{1,m}$, and $S_n^m$ the graph obtained from $K_{1,n}$ by subdividing $m$ edges once. Alon et al.\ conjectured that $r_\ell(K_{1,n};k)=r(K_{1,n};k)$ for all $k,n\ge1$. In this paper, we confirm their conjecture for all $k\ge1$ and $n\ge3$ by a unified direct proof. For even $k\ge4$ and under explicit parameter conditions, we prove that $r(S(n,m);k)=kn+m+2$ for even $n$ and $r(S_n^m;k)=k(n-1)+m+2$ for odd $n$. For two colors, we establish a general list Ramsey lower bound and determine the common values of Ramsey and list Ramsey numbers for double and subdivided stars in explicit parameter ranges. These results close several gaps in the known bounds.

math.CO↗

arXiv2026

Gap spectra and densities of slow Fibonacci walks

Let $F_1=F_2=1$ and $F_{t+2}=F_{t+1}+F_t$ for $t\geq1$. For every $n\geq2$, there are unique integers $a,b,t$ such that $n=aF_t+bF_{t-1}$ with $t\geq2$ and $1\leq a\leq b\leq F_t$. The Fibonacci walk with initial pair $(b,a)$ reaches $n$ as late as possible, and the term following $n$ in this walk is $\lfloorϕn\rfloor$ when $t$ is even and $\lceilϕn\rceil$ when $t$ is odd, where $ϕ=(1+\sqrt5)/2$. Let $D=\{d_1<d_2<\cdots\}$ and $U=\{u_1<u_2<\cdots\}$ be the sets corresponding to even and odd $t$, respectively. For $\ell,m\geq1$, define $D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$, $U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$, $D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$ and $U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$. Chung, Graham and Spiro conjectured that $D_\ell=U_\ell$ for all $\ell$, and asked for the densities of $D_\ell(m)$ and $U_\ell(m)$, especially when $\ell=1$. In this paper, we determine the third and fourth order gap spectra, and show that the conjecture holds for $\ell=3$ but fails for $\ell=4$. We also answer their density question by characterizing when $D_\ell(m)$ and $U_\ell(m)$ have natural densities and proving that their logarithmic densities always exist and are equal. For $\ell=1$, we give the exact logarithmic densities.

math.NT↗

arXiv2026

Ramsey multiplicity for ordered graphs

Let $\cG_1,\ldots,\cG_k$ be fixed vertex-ordered graphs, each containing at least one edge. The ordered Ramsey number $\oR(\cG_1,\ldots,\cG_k)$ is the least integer $N$ such that every $k$-edge-coloring of the ordered complete graph $\cK_N$ contains an order-preserving copy of $\cG_i$ in color $i$ for some $i\in[k]$. For positive weights $\blambda=(λ_1,\ldots,λ_k)$, let $\oM_{\blambda}(n;\cG_1,\ldots,\cG_k)$ denote the minimum weighted number of correctly colored, order-preserving copies of the target graphs over all $k$-edge-colorings of $\cK_n$. When $\blambda=\bf{1}$, $\oM_{\bf{1}}(n;\cG_1,\ldots,\cG_k)=\oM(n;\cG_1,\ldots,\cG_k)$ is called the ordered Ramsey multiplicity. In this paper, we first establish the amplification inequality \[ \oM_{\blambda}(n;\cG_1,\ldots,\cG_k) \ge \oM_{\blambda}(t;\cG_1,\ldots,\cG_k) \frac{\binom{n}{\hmin}}{\binom{t}{\hmin}}, \] where $h_i=v(\cG_i),\hmin=\min_{i\in[k]}h_i$, and $n\ge t\ge\oR(\cG_1,\ldots,\cG_k)$. Let $\cS_{r,s}$ be the ordered star whose center has $r-1$ leaves to its left and $s-1$ leaves to its right, and let $\bB_m$ be the family of all ordered perfect matchings on $[2m]$ containing the edge $\{1,2m\}$. We apply the amplification inequality to obtain the multiplicity lower bounds for ordered stars and ordered perfect matchings. We then obtain the upper bound $\oM_{\boldsymbolλ} (n;\cS_{r_1,s_1},\cS_{r_2,s_2}) \le \min\{λ_1 B_{h_1}(n),λ_2 B_{h_2}(n)\}$ by constructions, where $B_{h_i}(n):= \binom{\lfloor n/2\rfloor}{h_i} + \binom{\lceil n/2\rceil}{h_i}$ and $h_i=r_i+s_i-1$ for $i\in [2]$. We also derive a random-coloring upper bound for ordered stars and prove \[\oM(n; \bB_m,\bB_m) \le \binom{n}{2m} \frac{(2m-2)!}{2^{2m-2}(m-1)!}.\] Finally, we establish a regularity-based lifting theorem for ordered colorings.

math.CO↗

arXiv2026

SAMe: A Semantic Anatomy Mapping Engine for Robotic Ultrasound

Robotic ultrasound has advanced local image-driven control, contact regulation, and view optimization, yet current systems lack the anatomical understanding needed to determine what to scan, where to begin, and how to adapt to individual patient anatomy. These gaps make systems still reliant on expert intervention to initiate scanning. Here we present SAMe, a semantic anatomy mapping engine that provides robotic ultrasound with an explicit anatomical prior layer. SAMe addresses scan initiation as a target-to-anatomy-to-action process: it grounds under-specified clinical complaints into structured target organs, instantiates a patient-specific anatomical representation for the grounded targets from a single external body image, and translates this representation into control-facing 6-DoF probe initialization states without any additional registration using preoperative CT or MRI. The anatomical representation maintained by SAMe is explicit, lightweight (single-organ inference in 0.08s), and compatible with downstream control by design. Across semantic grounding, anatomical instantiation, and real-robot evaluation, SAMe shows strong performance across the full initialization pipeline. In real-robot experiments, centroid-based SAMe initialization outperformed the body-keypoint-based heuristic baseline under a budget-matched single-target setting for both liver (86.7% versus 46.7%) and kidney (80.0% versus 73.3%) initialization. Furthermore, The trial-level organ-hit rate reached 97.3% for liver and 83.3% for kidney when multiple candidate targets were available. These results establish an explicit anatomical prior layer that addresses scan initialization and is designed to support broader downstream autonomous scanning pipelines, providing the anatomical foundation for complaint-driven, anatomically informed robotic ultrasonography.

cs.CV↗

arXiv2022

Some exact values on Ramsey numbers related to fans

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that any red-blue edge-coloring of the complete graph $K_N$ contains a red $F$ or a blue $H$. When $F=H$, we simply write $R_2(H)$. For an positive integer $n$, let $K_{1,n}$ be a star with $n+1$ vertices, $F_n$ be a fan with $2n+1$ vertices consisting of $n$ triangles sharing one common vertex, and $nK_3$ be a graph with $3n$ vertices obtained from the disjoint union of $n$ triangles. In 1975, Burr, Erdős and Spencer \cite{B} proved that $R_2(nK_3)=5n$ for $n\ge2$. However, determining the exact value of $R_2(F_n)$ is notoriously difficult. So far, only $R_2(F_2)=9$ has been proved. Notice that both $F_n$ and $nK_3$ contain $n$ triangles and $|V(F_n)|<|V(nK_3)|$ for all $n\ge 2$. Chen, Yu and Zhao (2021) speculated that $R_2(F_n)\le R_2(nK_3)=5n$ for $n$ sufficiently large. In this paper, we first prove that $R(K_{1,n},F_n)=3n-\varepsilon$ for $n\ge1$, where $\varepsilon=0$ if $n$ is odd and $\varepsilon=1$ if $n$ is even. Applying the exact values of $R(K_{1,n},F_n)$, we will confirm $R_2(F_n)\le 5n$ for $n=3$ by showing that $R_2(F_3)=14$.

math.CO↗

arXiv2022

Bonds intersecting long paths in $k$-connected graphs

A well-known question of Gallai (1966) asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs, the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex $v$ if and only if it meets the vertex-bond with respect to $v$. Therefore, a very natural approach to Gallai's question is to study whether there is a bond meeting all longest paths. Let $p$ denote the length of a longest path of connected graphs. We show that for any 2-connected graph, there is a bond meeting all paths of length at least $p-1$. We then prove that for any 3-connected graph, there is a bond meeting all paths of length at least $p-2$. For a $k$-connected graph $(k\ge3)$, we show that there is a bond meeting all paths of length at least $p-t+1$, where $t=\Big\lfloor\sqrt{\frac{k-2}{2}}\Big\rfloor$ if $p$ is even and $t=\Big\lceil\sqrt{\frac{k-2}{2}}\Big\rceil$ if $p$ is odd. Our results provide analogs of the corresponding results of P. Wu and S. McGuinness [Bonds intersecting cycles in a graph, Combinatorica 25 (4) (2005), 439-450] also.

math.CO↗

arXiv2021

Gallai-Ramsey numbers for graphs with chromatic number three

Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we study Gallai-Ramsey numbers for graphs with chromatic number three such as $\widehat{K}_m$ for $m\ge2$, where $\widehat{K}_m$ is a kipas with $m+1$ vertices obtained from the join of $K_1$ and $P_m$, and a class of graphs with five vertices, denoted by $\mathscr{H}$. We first study the general lower bound of such graphs and propose a conjecture for the exact value of $GR_k(\widehat{K}_m)$. Then we give a unified proof to determine the Gallai-Ramsey numbers for many graphs in $\mathscr{H}$ and obtain the exact value of $GR_k(\widehat{K}_4)$ for $k\ge1$. Our outcomes not only indicate that the conjecture on $GR_k(\widehat{K}_m)$ is true for $m\le4$, but also imply several results on $GR_k(H)$ for some $H\in \mathscr{H}$ which are proved individually in different papers.

math.CO↗

arXiv2020

Gallai-Ramsey numbers for graphs with five vertices of chromatic number four

Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $H$. In this paper, we determine the Gallai-Ramsey numbers for connected graphs with five vertices of chromatic number four.

math.CO↗

arXiv2019

A note on Gallai-Ramsey number of even wheels

A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $W_{2n} $ denote an even wheel on $2n+1\ge5$ vertices. In this note, we study Gallai-Ramsey number of $W_{2n}$ and completely determine the exact value of $GR_k(W_4)$ for all $k\ge2$.

math.CO↗