arXiv subjects

Junying Lu

Publications and source records attributed to Junying Lu.

7 recordsLinked to original sources

arXiv2026

Threshold Ramsey multiplicity and extremal colorings for odd cycles

The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $n$ such that every red/blue edge-coloring of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ of $H$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_{r(H)}$. The only family for which $m(H)$ has been determined is stars, due to Harary and Prins (1974). Let $C_k$ be a cycle on $k$ vertices. Conlon, Fox, Sudakov, and Wei (2022) conjectured that $m(C_k)=(k-1)!/2$ for every sufficiently large odd integer $k$. In this paper, we confirm the conjecture and characterize all the extremal colorings of $K_{2k-1}$. This is also the second family for which $m(H)$ has been determined.

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arXiv2026

Proofs of two conjectures on generalizations of Brouwer's Laplacian conjecture

Let $G=(V,E)$ be a simple graph of order $n$ and let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ be the eigenvalues of its Laplacian matrix. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le |E|+\binom{k+1}{2}$, which was recently confirmed by Kothari and Tudose. Before Brouwer's conjecture was proved, Lew (JCT-B, 2026) established a weaker form of Brouwer's Laplacian eigenvalue inequality and proposed two conjectures for upper bounds on the sum of the $k$ largest Laplacian eigenvalues, one in terms of the matching number and the other in terms of the vertex-cover number. Using Brouwer's Laplacian inequality, we prove both conjectures.

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arXiv2026

Two-block cycles and chromatic number of Hamiltonian digraphs

Let $k$ and $\ell$ be positive integers. The family $C(k,\ell)$ consists of all digraphs obtained from two internally vertex-disjoint directed paths of lengths at least $k$ and $\ell$, respectively, and identifying their initial vertices and their terminal vertices. Addario-Berry, Havet and Thomass\'e (JCT-B, 2007) asked whether, for any positive integers $k$ and $\ell$ with $k+\ell \ge 4$, the chromatic number $\chi(D)$ is at most $k+\ell-1$ for every $C(k,\ell)$-free strongly connected digraph $D$. Let $D$ be a $C(k,\ell)$-free Hamiltonian digraph. Kim, Kim, Ma and Park (JGT, 2018) showed that $\chi(D) \le k+\ell$ and the bound is attained when $k+\ell=5$. In this paper, we prove that $\chi(D) \le k+\ell-1$ for $k+\ell\ge 6$ and this bound is best possible for all $k+\ell\geq 6$, which resolves the problem posed by Addario-Berry, Havet and Thomass\'e for Hamiltonian digraphs.

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arXiv2025

Oriented trees in digraphs with large girth

The girth of a graph $G$ is the length of a shortest cycle of $G$. Jiang (JCT-B, 2001) showed that every graph $G$ with girth at least $2\ell+1$ and minimum degree at least $k/\ell$ contains every tree $T$ with $k$ edges whose maximum degree does not exceed the minimum degree of $G$. Let $\delta^0(D)$ be the minimum semidegree of a digraph $D$ and $\Delta(D)$ be the maximum degree of $D$. In this paper, we establish a digraph version of Jiang's result, stating that every oriented graph $D$ of girth at least $2\ell+1$ with $\delta^0(D)\ge \max\{k/\ell,\Delta(T)\}$ contains every oriented tree with $k$ edges, that answers a question raised by Stein and Trujillo-Negrete in affirmative.

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arXiv2024

Oriented Ramsey numbers of some sparse graphs

Let $H$ be an oriented graph without directed cycle. The oriented Ramsey number of $H$, denoted by $\overrightarrow{r}(H)$, is the smallest integer $N$ such that every tournament on $N$ vertices contains a copy of $H$. Rosenfeld (JCT-B, 1974) conjectured that $\overrightarrow{r}(H)=|H|$ if $H$ is a cycle of sufficiently large order, which was confirmed for $|H|\geq 9$ by Zein recently, and so does if $H$ is a path. Note that $\overrightarrow{r}(H)=|H|$ implies any tournament contains $H$ as a spanning subdigraph, it is interesting to ask when $\overrightarrow{r}(H)=|H|$ for $H$ being a sparse oriented graph. S\'os (1986) conjectured this is true if $H$ is a directed path plus an additional edge containing the origin of the path as one end, which was confirmed by Petrovi\'{c} (JGT, 1988). In this paper, we show that $\overrightarrow{r}(H)=|H|$ for $H$ being an oriented graph obtained by identifying a vertex of an antidirected cycle with one end of a directed path. Some other oriented Ramsey numbers for oriented graphs with one cycle are also discussed.

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arXiv2023

Connectedness of friends-and-strangers graphs of complete bipartite graphs and others

Let $X$ and $Y$ be any two graphs of order $n$. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $σ:V(X) \mapsto V(Y)$, in which two bijections $σ$, $σ'$ are adjacent if and only if they differ precisely on two adjacent vertices of $X$, and the corresponding mappings are adjacent in $Y$. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected. Let $K_{k,n-k}$ be a complete bipartite graph of order $n$. In 1974, Wilson characterized the connectedness of $\mathsf{FS}(K_{1,n-1},Y)$ by using algebraic methods. In this paper, by using combinatorial methods, we investigate the connectedness of $\mathsf{FS}(K_{k,n-k},Y)$ for any $Y$ and all $k\ge 2$, including $Y$ being a random graph, as suggested by Defant and Kravitz, and pose some open problems.

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arXiv2022

Spectral radius of graphs forbidden $C_7$ or $C_6^{\triangle}$

Let $C_k^{\triangle}$ be the graph obtained from a cycle $C_{k}$ by adding a new vertex connecting two adjacent vertices in $C_{k}$. In this note, we obtain the graph maximizing the spectral radius among all graphs with size $m$ and containing no subgraph isomorphic to $C_6^{\triangle}$. As a byproduct, we will show that if the spectral radius $\lambda(G)\ge1+\sqrt{m-2}$, then $G$ must contains all the cycles $C_i$ for $3\le i\le 7$ unless $G\cong K_3\nabla \left(\frac{m-3}{3}K_1\right)$.

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