arXiv subjects

Yaping Mao

Publications and source records attributed to Yaping Mao.

At least 55 records · Page 3Linked to original sources

arXiv2023

Complete bipartite graphs without small rainbow stars

The $k$-edge-colored bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$ is defined as the minimum integer $n$ such that $n^2\geq k$ and for every $N\geq n$, every edge-coloring (using all $k$ colors) of complete bipartite graph $K_{N,N}$ contains a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we first study the structural theorem on the complete bipartite graph $K_{n,n}$ with no rainbow copy of $K_{1,3}$. Next, we utilize the results to prove the exact values of $\operatorname{bgr}_{k}(P_4: H)$, $\operatorname{bgr}_{k}(P_5: H)$, $\operatorname{bgr}_{k}(K_{1,3}: H)$, where $H$ is a various union of cycles and paths and stars.

math.CO↗

arXiv2023

Streaming Algorithms for the $k$-Submodular Cover Problem

Given a natural number $k\ge 2$, we consider the $k$-submodular cover problem ($k$-SC). The objective is to find a minimum cost subset of a ground set $\mathcal{X}$ subject to the value of a $k$-submodular utility function being at least a certain predetermined value $τ$. For this problem, we design a bicriteria algorithm with a cost at most $O(1/ε)$ times the optimal value, while the utility is at least $(1-ε)τ/r$, where $r$ depends on the monotonicity of $g$.

cs.DS↗

arXiv2023

Note on the connectivity keeping spiders in $k$-connected graphs

W. Mader [J. Graph Theory 65 (2010), 61--69] conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with $δ(G)\geq\lfloor\frac{3k}{2}\rfloor+m-1$ contains a tree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2010, Mader confirmed the conjecture for the $k$-connected graph if $T$ is a path; very recently, Liu et al. confirmed the conjecture if $k=2,3$. The conjecture is open for $k\geq 4$ till now. In this paper, we show that Mader's conjecture is true for the $k+1$-connected graph if $T$ is a spider and $Δ(G)=|G|-1$.

math.CO↗

arXiv2023

Ramsey Achievement Games on Graphs : Algorithms and Bounds

In 1982, Harary introduced the concept of Ramsey achievement game on graphs. Given a graph $F$ with no isolated vertices. Consider the following game played on the complete graph $K_n$ by two players Alice and Bob. First, Alice colors one of the edges of $K_n$ blue, then Bob colors a different edge red, and so on. The first player who can complete the formation of $F$ in his color is the winner. The minimum $n$ for which Alice has a winning strategy is the achievement number of $F$, denoted by $a(F)$. If we replace $K_n$ in the game by the completed bipartite graph $K_{n,n}$, we get the bipartite achievement number, denoted by $\operatorname{ba}(F)$. In his seminal paper, Harary proposed an open problem of determining bipartite achievement numbers for trees. In this paper, we correct $\operatorname{ba}(mK_2)=m+1$ to $m$ and disprove $\operatorname{ba}(K_{1,m})=2m-2$ from Erickson and Harary, and extend their results on bipartite achievement numbers. We also find the exact values of achievement numbers for matchings, and the exact values or upper and lower bounds of bipartite achievement numbers on matchings, stars, and double stars. Our upper bounds are obtained by deriving efficient winning strategies for Alice.

math.CO↗

arXiv2023

Gallai-Ramsey Multiplicity

Given two graphs $G$ and $H$, the \emph{general $k$-colored Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined to be the minimum integer $m$ such that every $k$-coloring of the complete graph on $m$ vertices contains either a rainbow copy of $G$ or a monochromatic copy of $H$. Interesting problems arise when one asks how many such rainbow copy of $G$ and monochromatic copy of $H$ must occur. The \emph{Gallai-Ramsey multiplicity} $\operatorname{GM}_{k}(G,H)$ is defined as the minimum total number of rainbow copy of $G$ and monochromatic copy of $H$ in any exact $k$-coloring of $K_{\operatorname{gr}_{k}(G,H)}$. In this paper, we give upper and lower bounds for Gallai-Ramsey multiplicity involving some small rainbow subgraphs.

math.CO↗

arXiv2023

Bounds for Gallai-Ramsey functions and numbers

For two graphs $G,H$ and a positive integer $k$, the \emph{Gallai-Ramsey number} $\operatorname{gr}_k(G,H)$ is defined as the minimum number of vertices $n$ such that any $k$-edge-coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $H$. If $G$ and $H$ are both complete graphs, then we call it Gallai-Ramsey function. Fox and Sudakov proved $\operatorname{gr}_k(K_s,K_t)\leq s^{4kt}$. Alon et al. showed that $\operatorname{gr}_k(K_s,K_t)\leq (2s^3+4s^2)^{kt}$. In this paper, we prove that $\operatorname{gr}_k(K_s,K_t)\leq 2^{kt}s^{3kt}$ for $t\geq 47$. We also give better upper bounds for $\operatorname{gr}_k(G,H)$ when $G,H$ are some special graphs. In this paper, we derive some lower bounds for Gallai-Ramsey functions and numbers by Lovász Local Lemma.

math.CO↗

arXiv2022

On the distance-edge-monitoring numbers of graphs

Foucaud et al. [Discrete Appl. Math. 319 (2022), 424-438] recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. For a set $M$ of vertices and an edge $e$ of a graph $G$, let $P(M, e)$ be the set of pairs $(x, y)$ with a vertex $x$ of $M$ and a vertex $y$ of $V(G)$ such that $d_G(x, y)\neq d_{G-e}(x, y)$. For a vertex $x$, let $EM(x)$ be the set of edges $e$ such that there exists a vertex $v$ in $G$ with $(x, v) \in P(\{x\}, e)$. A set $M$ of vertices of a graph $G$ is 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 distance-edge-monitoring number of a graph $G$, denoted by $dem(G)$, is defined as the smallest size of distance-edge-monitoring sets of $G$. The vertices of $M$ represent distance probes in a network modeled by $G$; when the edge $e$ fails, the distance from $x$ to $y$ increases, and thus we are able to detect the failure. It turns out that not only we can detect it, but we can even correctly locate the failing edge. In this paper, we continue the study of \emph{distance-edge-monitoring sets}. In particular, we give upper and lower bounds of $P(M,e)$, $EM(x)$, $dem(G)$, respectively, and extremal graphs attaining the bounds are characterized. We also characterize the graphs with $dem(G)=3$.

math.CO↗

arXiv2022

Euclidean Gallai-Ramsey Theory

In this paper, we introduce Euclidean Gallai-Ramsey theory, by combining Euclidean Ramsey theory and Gallai-Ramsey theory on graphs. More precisely, we consider the following problem: For an integer $r$ and configurations $K$ and $K'$, does there exist an integer $n_0$ such that for any $r$-coloring of the points of $n$-dimensional Euclidean space with $n \geq n_0$, there is a monochromatic configuration congruent to $K$ or a rainbow configuration congruent to $K'$? In particular, we give a bound on $n_0$ for some configurations $K$ and $K'$, such as triangles and rectangles. Those are extensions of ordinary Euclidean Ramsey theory where the purpose is to find a monochromatic configuration.

math.CO↗

arXiv2022

A general approach to deriving diagnosability results of interconnection networks

We generalize an approach to deriving diagnosability results of various interconnection networks in terms of the popular $g$-good-neighbor and $g$-extra fault-tolerant models, as well as mainstream diagnostic models such as the PMC and the MM* models. As demonstrative examples, we show how to follow this constructive, and effective, process to derive the $g$-extra diagnosabilities of the hypercube, the $(n, k)$-star, and the arrangement graph. These results agree with those achieved individually, without duplicating structure independent technical details. Some of them come with a larger applicable range than those already known, and the result for the arrangement graph in terms of the MM* model is new.

cs.NI↗

arXiv2022

Some multivariable Rado numbers

The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let $\mathcal{E}$ be a linear equation. Denote by $\operatorname{R}_r(\mathcal{E})$ the minimal integer, if it exists, such that any $r$-coloring of $[1,\operatorname{R}_r(\mathcal{E})]$ must admit a monochromatic solution to $\mathcal{E}$. In this paper, we give upper and lower bounds for the Rado number of $\sum_{i=1}^{m-2}x_i+kx_{m-1}=\ell x_{m}$, and some exact values are also given. Furthermore, we derive some results for the cases that $\ell=m=4$ and $m=5, \ell=k+i \ (1\leq i\leq 5)$. As a generalization, the \emph{$r$-color Rado numbers} for linear equations $\mathcal{E}_1,\mathcal{E}_2,...,\mathcal{E}_r$ is defined as the minimal integer, if it exists, such that any $r$-coloring of $[1,\operatorname{R}_r(\mathcal{E}_1,\mathcal{E}_2,...,\mathcal{E}_r)]$ must admit a monochromatic solution to some $\mathcal{E}_i$, where $1\leq i\leq r$. A lower bound for $\operatorname{R}_r(\mathcal{E}_1,\mathcal{E}_2,...,\mathcal{E}_r)$ and the exact values of $\operatorname{R}_2(x+y=z,\ell x=y)=5k$ and $\operatorname{R}_2(x+y=z, x+a=y)$ was given by Lovász Local Lemma.

math.CO↗

arXiv2021

Gallai-Ramsey numbers involving a rainbow $4$-path

Given two non-empty graphs $G,H$ and a positive integer $k$, the Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains either a rainbow colored copy of $G$ or a monochromatic copy of $H$. In this paper, we got some exact values or bounds for $\operatorname{gr}_k(P_5:H) \ (k\geq 3)$ if $H$ is a general graph or a star with extra independent edges or a pineapple.

math.CO↗

arXiv2021

Nordhaus-Guddum type results for the Steiner Gutman index of graphs

Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The \emph{Steiner Gutman $k$-index} $\operatorname{SGut}_k(G)$ of $G$ is defined by $\operatorname{SGut}_k(G)$ $=\sum_{S\subseteq V(G), \ |S|=k}\left(\prod_{v\in S}deg_G(v)\right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $\operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $\operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $\operatorname{SGut}_k(G)+\operatorname{SGut}_k(\overline{G})$ and $\operatorname{SGut}_k(G)\cdot \operatorname{SGut}_k(\overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $Δ$, minimum degree $δ$.

math.CO↗

arXiv2021

Gallai-Ramsey numbers of odd cycles

Given two graphs $G$ and $H$ and a positive integer $k$, the $k$-color Gallai-Ramsey number, denoted by $gr_{k}(G : H)$, is the minimum integer $N$ such that for all $n \geq N$, every $k$-coloring of the edges of $K_{n}$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. We prove that $gr_{k} (K_{3} : C_{2\ell + 1}) = \ell \cdot 2^{k} + 1$ for all $k \geq 1$ and $\ell \geq 3$.

math.CO↗

arXiv2020

Gallai-Ramsey number for the union of stars

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain the exact value of the Gallai-Ramsey numbers for the union of two stars in many cases and bounds in other cases. This work represents the first class of disconnected graphs to be considered as the desired monochromatic subgraph.

math.CO↗

arXiv2020

Gallai Ramsey number for double stars

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for double stars $S(n,m)$, where $S(n,m)$ is the graph obtained from the union of two stars $K_{1,n}$ and $K_{1,m}$ by adding an edge between their centers. We also provide the sharp result in some cases.

math.CO↗

arXiv2019

Fractional matching preclusion number of graphs

The \emph{fractional matching preclusion number} of a graph $G$, denoted by $fmp(G)$, is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings. In this paper, we first give some sharp upper and lower bounds of fractional matching preclusion number. Next, graphs with large and small fractional matching preclusion number are characterized, respectively. In the end, we investigate some extremal problems on fractional matching preclusion number.

math.CO↗

arXiv2019

Steiner 3-diameter, maximum degree and size of a graph

The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erdös and Rényi. In this paper, we generalize the above problem for Steiner $k$-diameter, and study the problem of determining the minimum size of a graph of order $n$ whose Steiner $3$-diameter is at most $d$ and whose maximum is at most $\ell$.

math.CO↗

arXiv2019

Fractional matching preclusion for generalized augmented cubes

The \emph{matching preclusion number} of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu recently introduced the concept of fractional matching preclusion number. The \emph{fractional matching preclusion number} of $G$ is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The \emph{fractional strong matching preclusion number} of $G$ is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional matching preclusion number and the fractional strong matching preclusion number for generalized augmented cubes. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.

math.CO↗