arXiv subjects

Yaping Mao

Publications and source records attributed to Yaping Mao.

At least 37 records · Page 2Linked to original sources

arXiv2026

Asymptotic Bounds for t(3,n) and an Application to t(4,n)

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ that is adjacent to $x$ and to no other vertex of $X$. The \emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in the blue subgraph or an $n$-element independent set in the red subgraph. The irredundant Ramsey number $s(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ has an $m$-element irredundant set in the blue subgraph or an $n$-element irredundant set in the blue subgraph. In this paper, we determine $t(3,n)$ and $s(3,n)$ up to a constant factor by showing that $t(3,n)=O\left(n^{5/4}/{\log{n}}\right)$, which improved the best upper bound due to Rousseau and Speed in [Comb. Probab. Comput. 12 (2003), 653-660]. As an application, we verify a conjecture for $m=4$ proposed by Chen, Hattingh, and Rousseau in [J. Graph Theory 17(2) (1993), 193-206].

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arXiv2026

Counterexamples to the Corsten-Frankl conjecture on diameter-Ramsey simplices

Corsten and Frankl conjectured that a simplex is diameter-Ramsey if and only if its circumcenter lies in its convex hull. We disprove this conjecture in every dimension $d\ge 3$. The main tool is a sufficient criterion based on a higher-order deficit decomposition: if the squared deficits $D^2-\|p_i-p_j\|^2$ admit a nonnegative decomposition over subsets of the vertex set, with total mass at most $D^2$, then the simplex is diameter-Ramsey. The pairwise deficit criterion of Frankl--Pach--Reiher--Rödl is recovered as a special case. As an application, for every $d\ge 3$ we construct a diameter-Ramsey $d$-simplex whose circumcenter lies outside its convex hull. A particularly simple family has squared edge lengths $\|p_1-p_2\|^2=\|p_1-p_j\|^2=7~ (4\le j\le d+1)$, $\|p_1-p_3\|^2=4$, and $\|p_i-p_j\|^2=4 ~ (2\le i<j\le d+1)$.

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arXiv2026

The Erdős-Faudree Problems and the Isolate-Free Core

In 1981, Erdős and Faudree asked whether there exists an infinite family of graphs $G_N$ on $N$ vertices with $Δ(G_N)<N-1$ and $\sri(G_N)=1$, and whether every family with $|V(G_N)|=N$ and $Δ(G_N)<c$ for some fixed constant $c$ must satisfy $\sri(G_N)\to 0$. We show first that the literal forms of the two questions are controlled entirely by isolated vertices: for every nonempty graph $G$, the whole sequence $\bigl(\sr(tK_2,G)\bigr)_{t\ge 1}$ depends only on the isolate-free core $\core(G)$. Consequently, Problem 1 has a positive answer and Problem~2 has a negative answer in exactly their original form. We then turn to the genuine content behind the two problems. For Problem 1 we study connected graphs and prove a complete limit theorem: for every $α\in[0,1]$ there exists a family of connected bipartite graphs $G_N$ with $|V(G_N)|=N$ and $\sri(G_N)\toα$; in particular there are connected graphs with $Δ(G_N)=N-2$ and $\sri(G_N)\to 1$. For Problem~2 we prove a strengthened positive statement: if $Δ(G_N)<c$ for a fixed constant $c$ and the isolate-free core of $G_N$ has order tending to infinity, then $\sri(G_N)\to 0$. In particular every connected bounded-degree family satisfies $\sri(G_N)\to 0$. Thus the original Erdős-Faudree questions are resolved in their literal form, and the mechanism behind their connected and disconnected behavior is identified precisely.

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arXiv2026

On two conjectures for generalized off-diagonal Schur numbers

For an integer $t \geq 3$, let $\mathcal{L}(t)$ denote the linear equation $x_1 + x_2 + \cdots + x_{t-1} = x_t,$ where all variables are positive integers. For integers $k \geq 1$ and $t_0,t_1,\dots,t_{k-1} \geq 3$, the generalized Schur number $S(k;t_0,t_1,\dots,t_{k-1})$ is the least positive integer $N$ such that every $k$-coloring of $[1,N]$, for some $i \in \{0,1,\dots,k-1\}$, a solution to $\mathcal{L}(t_i)$ with all variables monochromatic in color $i$. In 2015, Ahmed and Schaal proposed a conjecture: $S(3 ; 3, t, u)>3 t u-t u-u-1$ for $3=t<u$ and $3<t \leq u$. In this paper, we confirm this conjecture. At the same paper, they also conjecture that $S(3 ; s, t, u)=s t u-t u-u-1$ for $4 \leq s \leq t \leq u$. Motivated by the second conjecture, we give a recursive lower bound of $S(r; k_0, k_1, \dots, k_{r-1})$ and upper bounds for $S(r; k_0, k_1, \dots, k_{r-1})$ and $S(r;k_0,\dots,k_{r-2},u)$ for all sufficiently large $u$.

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arXiv2026

On the minimum number of monochromatic solutions to the strict Schur inequality in 2-colored integer intervals with negative left endpoint

Kosek, Robertson, Sabo, and Schaal studied the minimum number $M_k(n)$ of monochromatic solutions to the strict Schur inequality system $x_1\le x_2\le x_3$ and $x_1+x_2<x_3$ in $2$-colorings of $[k+1,k+n]$. They proved that for every fixed $k\ge 0$, $M_k(n)= \frac{n^3}{12(1+2\sqrt2)^2}(1+o_k(1)),$ and left open the case $k\le -2$. In this paper, we resolve that remaining range.

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arXiv2026

Induced Ramsey numbers for fans

The induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is defined as the minimum order of a graph $F$ on such that any 2-coloring of its edges with red and blue leads to either a red induced copy of $G$ or a blue induced copy of $H$. Motivated by the Kohayakawa-Prömel-Rödl conjecture, we prove that a quadratic upper bound $\mathrm{r}_{\text {ind}}\left(G, F_n\right) \leq C n^2$ for fixed $G$, where $F_n$ is a graph with one central vertex, $2n$ leaf vertices, and $n$ disjoint edges. In particular, for star graphs $K_{1, \ell}$ $(\ell \leq n)$, constructive coloring and matching arguments yield $2 n+2 \ell-1 \leq \mathrm{r}_{\text {ind}}\left(K_{1, \ell}, F_n\right) \leq(\ell+n-1)(\ell+1)+1$, with the exact value $\mathrm{r}_{\text {ind}}\left(K_{1,2}, F_n\right)=3 n+4$.

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arXiv2026

Boolean lattice without small rainbow subposets

A Boolean lattice $\mathcal{B}_n=(2^X, \leq)$ is the power set of an $n$-element ground set $X$ equipped with inclusion relation. For two posets $\mathcal{P}$ and $\mathcal{Q}$, we say that $\mathcal{Q}$ contains an \emph{induced copy} of $\mathcal{P}$ if there exists an injection $f : \mathcal{P} \to \mathcal{Q}$ such that $f(X) \le f(Y)$ if and only if $X \le Y$ in $\mathcal{P}$. A $k$-coloring is exact if all colors are used at least once. For posets $\mathcal{Q}$ and $\mathcal{P}$, the \emph{Boolean Gallai-Ramsey number} $\operatorname{GR}_{k}(\mathcal{Q}:\mathcal{P})$ is defined as the smallest $n$ such that any exact $k$-coloring of the sets in $\mathcal{B}_n$ contains either a rainbow induced copy of $\mathcal{Q}$ or a monochromatic induced copy of $\mathcal{P}$ and the \emph{Boolean rainbow Ramsey number} $\operatorname{RR}(\mathcal{Q}:\mathcal{P})$ is defined as the smallest $n$ such that any coloring of the sets in $\mathcal{B}_n$ contains either a rainbow induced copy of $\mathcal{Q}$ or a monochromatic induced copy of $\mathcal{P}$. In this paper, we first study the structural properties of exact $k$-colorings of the sets in Boolean lattice without rainbow induced copy of small posets. As the application of these results, we give exact values and some bounds of Boolean Gallai-Ramsey numbers and Boolean rainbow Ramsey numbers, which improve a result of Chen, Cheng, Li, and Liu in 2020 and give an answer of a question proposed by Chang, Gerbner, Li, Methuku, Nagy, Patkós, and Vizer in 2022.

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arXiv2025

On the size edge-ordered Ramsey numbers of graphs

For edge-ordered graphs $G^{\prec}$ and $H^{\prec}$, the size edge-ordered Ramsey number $\hat{r}_{\text{edge}}(G^{\prec}, H^{\prec})$ is defined as the smallest integer $m$ for which there exists an edge-ordered graph $F^{\prec}$ (with underlying graph $F$) having $m$ edges, such that every $2$-coloring of the edges of $F^{\prec}$ contains a monochromatic edge-ordered subgraph isomorphic to $G^{\prec}$ or a monochromatic edge-ordered subgraph isomorphic to $H^{\prec}$. Fox and Li posed a foundational question: which families of edge-ordered graphs have linear or near-linear size edge-ordered Ramsey numbers? In this paper, we apply Szemerédi's regularity lemma to prove that, even for sparse graph families, specifically the well-defined class of edge-ordered book graphs, the size edge-ordered Ramsey numbers of this family exhibit non-linear growth. Furthermore, we show that three families of edge-ordered graphs exhibit linear or near-linear size edge-ordered Ramsey numbers.

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arXiv2025

Ramsey numbers for partially-ordered sets

We say that a poset $Q$ contains a copy (resp.~an induced copy) of a poset $P$ if there is an injection $f : P \to Q$ such that for any $x,y \in P$, $f(x)\leq f(y)$ in $Q$ if (resp.~if and only if) $x\leq y$ in $P$. Let $\mathcal{Q}=\{Q_{n} : n\geq 1\}$ be a family of posets such that $Q_n\subseteq Q_{n+1}$ and $|Q_n|<|Q_{n+1}|$ for each $n$. For given $k$ posets $P_1, P_2, \dots , P_k$, the \emph{weak (resp.~strong) poset Ramsey number for $t$-chains} is the smallest number $n$ such that for any coloring of $t$-chains in $Q_n\in \mathcal{Q}$ with $k$ colors, say $1,2, \dots, k$, there is a monochromatic (resp.~induced) copy of the poset $P_i$ in color $i$ for some $1\leq i\leq k$. In this paper, we give several lower and upper bounds on the weak and strong poset Ramsey number for $t$-chains.

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arXiv2025

Constructing disjoint Steiner trees in Sierpiński graphs

Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are \emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if $E(T_i) \cap E(T_j )=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for every pair of distinct integers $i,j$, $1 \leq i, j \leq \ell$. Similarly, if we only have the condition $E(T_i) \cap E(T_j )=\emptyset$ but without the condition $V(T_i)\cap V(T_j)=S$, then they are \emph{edge-disjoint Steiner trees}. The \emph{generalized $k$-connectivity}, denoted by $κ_k(G)$, of a graph $G$, is defined as $κ_k(G)=\min\{κ_G(S)|S \subseteq V(G) \ \textrm{and} \ |S|=k \}$, where $κ_G(S)$ is the maximum number of internally disjoint $S$-Steiner trees. The \emph{generalized local edge-connectivity} $λ_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$. The {\it generalized $k$-edge-connectivity} $λ_k(G)$ of $G$ is defined as $λ_k(G)=\min\{λ_{G}(S)\,|\,S\subseteq V(G) \ and \ |S|=k\}$. These measures are generalizations of the concepts of connectivity and edge-connectivity, and they and can be used as measures of vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs. In this paper, we obtain the exact value of $λ_{k}(S(n,\ell))$ for $3\leq k\leq \ell^n$, and the exact value of $κ_{k}(S(n,\ell))$ for $3\leq k\leq \ell$, where $S(n, \ell)$ is the Sierpiński graphs with order $\ell^n$. As a direct consequence, these graphs provide additional interesting examples when $λ_{k}(S(n,\ell))=κ_{k}(S(n,\ell))$. We also study the some network properties of Sierpiński graphs.

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arXiv2025

On the set-coloring Ramsey numbers of graphs

The \textit{set-coloring Ramsey number} $\mathrm{R}_{r, s}(G_1,G_2,...,G_r)$ is the least $n \in \mathbb{N}$ such that every coloring $χ: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a monochromatic copy of $G_i$, that is, a color $i \in[r]$ such that $i \in χ(e)$ for every $e \in E(G_i)$. If $G_1=G_2=\cdots=G_r=G$, then we write $\mathrm{R}_{r,s}(G)$ for short. In 2022, Le asked to find lower and upper bounds for $\mathrm{R}_{s, t}(G)$ with various kinds of graphs $G$ such as stars, paths, cycles, etc. In this paper, we obtain exact values or bounds for the set-coloring Ramsey numbers of stars, paths, matchings, etc. By Lovász Local Lemma, we give a lower bound for the set-coloring Ramsey number for general graphs.

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arXiv2025

The $g$-good-neighbor diagnosability of product networks under the PMC model

The concept of neighbor connectivity originated from the assessment of the subversion of espionage networks caused by underground resistance movements, and it has now been applied to measure the disruption of networks caused by cascading failures through neighbors. In this paper, we give two necessary and sufficient conditions of the existance of $g$-good-neighbor diagnosability. We introduce a new concept called $g$-good neighbor cut-component number (gc number for short), which has close relation with $g$-good-neighbor diagnosability. Sharp lower and upper bounds of the gc number of general graphs in terms of the $g$-good neighbor connectivity is given, which provides a formula to compute the $g$-good-neighbor diagnosability for general graphs (therefore for Cartesian product graphs). As their applications, we get the exact values or bounds for the gc numbers and $g$-good-neighbor diagnosability of grid, torus networks and generalized cubes.

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arXiv2025

Gallai-Schur Triples and Related Problems

Schur's Theorem states that, for any $r \in \mathbb{Z}^+$, there exists a minimum integer $S(r)$ such that every $r$-coloring of $\{1,2,\dots,S(r)\}$ admits a monochromatic solution to $x+y=z$. Recently, Budden determined the related Gallai-Schur numbers; that is, he determined the minimum integer $GS(r)$ such that every $r$-coloring of $\{1,2,\dots,GS(r)\}$ admits either a rainbow or monochromatic solution to $x+y=z$. In this article we consider problems that have been solved in the monochromatic setting under a monochromatic-rainbow paradigm. In particular, we investigate Gallai-Schur numbers when $x \neq y$, we consider $x+y+b=z$ and $x+y<z$, and we investigate the asymptotic minimum number of rainbow and monochromatic solutions to $x+y=z$ and $x+y<z$.

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arXiv2024

On the two problems in Ramsey achievement games

Let $p,q$ be two integers with $p\geq q$. Given a finite graph $F$ with no isolated vertices, the generalized Ramsey achievement game of $F$ on the complete graph $K_n$, denoted by $(p,q;K_n,F,+)$, is played by two players called Alice and Bob. In each round, Alice firstly chooses $p$ uncolored edges $e_1,e_2,...,e_p$ and colors it blue, then Bob chooses $q$ uncolored edge $f_1,f_2,...,f_q$ and colors it red; the player who can first complete the formation of $F$ in his (or her) color is the winner. The generalized achievement number of $F$, denoted by ${a}(p,q;F)$ is defined to be the smallest $n$ for which Alice has a winning strategy. If $p=q=1$, then it is denoted by ${a}(F)$, which is the classical achievement number of $F$ introduced by Harary in 1982. If Alice aims to form a blue $F$, and the goal of Bob is to try to stop him, this kind of game is called the first player game by Bollobás. Let ${a}^*(F)$ be the smallest positive integer $n$ for which Alice has a winning strategy in the first player game. A conjecture due to Harary states that the minimum value of ${a}(T)$ is realized when $T$ is a path and the maximum value of ${a}(T)$ is realized when $T$ is a star among all trees $T$ of order $n$. He also asked which graphs $F$ satisfy $a^*(F)=a(F)$? In this paper, we proved that $n\leq {a}(p,q;T)\leq n+q\left\lfloor (n-2)/p \right\rfloor$ for all trees $T$ of order $n$, and obtained a lower bound of ${a}(p,q;K_{1,n-1})$, where $K_{1,n-1}$ is a star. We proved that the minimum value of ${a}(T)$ is realized when $T$ is a path which gives a positive solution to the first part of Harary's conjecture, and ${a}(T)\leq 2n-2$ for all trees of order $n$. We also proved that for $n\geq 3$, we have $2n-2-\sqrt{(4n-8)\ln (4n-4)}\leq a(K_{1,n-1})\leq 2n-2$ with the help of a theorem of Alon, Krivelevich, Spencer and Szabó. We proved that $a^*(P_n)=a(P_n)$ for a path $P_n$.

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arXiv2024

Perturbation results for distance-edge-monitoring numbers

Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Given a graph $G=(V(G), E(G))$, a set $M \subseteq V(G)$ is a distance-edge-monitoring set if for every edge $e \in E(G)$, there is a vertex $x \in M$ and a vertex $y \in V(G)$ such that the edge $e$ belongs to all shortest paths between $x$ and $y$. The smallest size of such a set in $G$ is denoted by $\operatorname{dem}(G)$. Denoted by $G-e$ (resp. $G \backslash u$) the subgraph of $G$ obtained by removing the edge $e$ from $G$ (resp. a vertex $u$ together with all its incident edges from $G$). In this paper, we first show that $\operatorname{dem}(G-e)- \operatorname{dem}(G)\leq 2$ for any graph $G$ and edge $e \in E(G)$. Moreover, the bound is sharp. Next, we construct two graphs $G$ and $H$ to show that $\operatorname{dem}(G)-\operatorname{dem}(G\setminus u)$ and $\operatorname{dem}(H\setminus v)-\operatorname{dem}(H)$ can be arbitrarily large, where $u \in V(G)$ and $v \in V(H)$. We also study the relation between $\operatorname{dem}(H)$ and $\operatorname{dem}(G)$, where $H$ is a subgraph of $G$. In the end, we give an algorithm to judge whether the distance-edge monitoring set still remain in the resulting graph when any edge of the graph $G$ is deleted.

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arXiv2024

Asymptotic Bounds for CO-irredundant and Irredundant Ramsey Numbers

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant (CO-irredundant) if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ ($y\in V$) that is adjacent to $x$ and to no other vertex of $X$. The irredundant Ramsey number $s(t_{1},\ldots,t_{l})$, CO-irredundant Ramsey number $s_{\operatorname{CO}}(t_{1},\ldots,t_{l})$, is the minimum $N$ such that every $l$-coloring of the edges of the complete graph $K_{N}$ on $N$ vertices has a monochromatic irredundant set, a monochromatic CO-irredundant set, of size $t_{i}$ for some $1\leq i\leq l$, respectively. In this paper, firstly, we establish a lower bound for the irredundant Ramsey number $s(t_{1},\ldots,t_{l})$ by a random and probabilistic method. Secondly, we improve an upper bound for $s(3,9)$ such that $24\leq s(3,9)\leq 26$. Thirdly, using Krivelevich's lemma, we establish an asymptotic lower bound for the $\operatorname{CO}$-irredundant Ramsey number $s_{\operatorname{CO}}(m,n)$.

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arXiv2024

Monitoring the edges of product networks using distances

Foucaud {\it et al.} recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Let $G$ be a graph with vertex set $V(G)$, $M$ a subset of $V(G)$, and $e$ be an edge in $E(G)$, and let $P(M, e)$ be the set of pairs $(x,y)$ such that $d_G(x, y)\neq d_{G-e}(x, y)$ where $x\in M$ and $y\in V(G)$. $M$ is called a \emph{distance-edge-monitoring set} if every edge $e$ of $G$ is monitored by some vertex of $M$, that is, the set $P(M, e)$ is nonempty. The {\em distance-edge-monitoring number} of $G$, denoted by $\operatorname{dem}(G)$, is defined as the smallest size of distance-edge-monitoring sets of $G$. For two graphs $G,H$ of order $m,n$, respectively, in this paper we prove that $\max\{m\operatorname{dem}(H),n\operatorname{dem}(G)\} \leq\operatorname{dem}(G\,\Box \,H) \leq m\operatorname{dem}(H)+n\operatorname{dem}(G) -\operatorname{dem}(G)\operatorname{dem}(H)$, where $\Box$ is the Cartesian product operation. Moreover, we characterize the graphs attaining the upper and lower bounds and show their applications on some known networks. We also obtain the distance-edge-monitoring numbers of join, corona, cluster, and some specific networks.

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arXiv2024

Ramsey and Gallai-Ramsey numbers for linear forests and kipas

For two graphs $G,H$, the \emph{Ramsey number} $r(G,H)$ is the minimum integer $n$ such that any red/blue edge-coloring of $K_n$ contains either a red copy of $G$ or a blue copy of $H$. For two graphs $G,H$, the \emph{Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $n$ such that any $k$-edge-coloring of $K_n$ must contain either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, the classical Ramsey numbers of linear forest versus kipas are obtained. We obtain the exact values of $\operatorname{gr}_k(G:H)$, where $H$ is either a path or a kipas and $G\in\{K_{1,3},P_4^+,P_5\}$ and $P_4^+$ is the graph consisting of $P_4$ with one extra edge incident with inner vertex.

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