arXiv subjects

Xinmin Hou

Publications and source records attributed to Xinmin Hou.

At least 37 records · Page 2Linked to original sources

arXiv2024

Ramsey-type results on parameters related to domination

The following inequality chain $$ ir(G)\le γ(G)\le i(G)\le α(G) \le Γ(G) \le I\!R(G)$$ is known as a domination chain, where $ir(G), γ(G), i(G), α(G), Γ(G)$, and $I\!R(G)$ are the lower irredundance number, the domination number, the independence domination number, the independence number, the upper domination number and the upper irredundance number of $G$, respectively. The Ramsey-type problem seeks to characterize the family $\mathcal H$ of graphs such that every $\mathcal H$-free graph $G$ has a bounded parameter $μ$. The classical Ramsey's theorem states that every $\{K_n, E_n\}$-free graph has a bounded number of vertices. Furuya (Discrete Math.Theor 2018) characterized $\mathcal H$ such that every connected $\mathcal H$-free graph $G$ has a bounded domination number. The characterization of the graph family $\mathcal H$ for which every connected $\mathcal H$-free graph $G$ has a bounded independence number was due to Choi, Furuya, Kim, Park~(Discrete math. 2020) and Chiba, Furuya (Electron. J. Combin., 2022). In this paper, we further characterize $\mathcal H$ such that every connected $\mathcal H$-free graph $G$ has bounded $μ(G)$ for $μ$ belonging to the set $\{ir(G), i(G), Γ(G), \text{IR}(G)\}$. This completes the characterization of $\mathcal H$ for which every connected $\mathcal H$-free graph $G$ has bounded $μ(G)$ for $μ(G)$ along the domination chain. Additionally, we characterize $\mathcal H$ such that every connected $\mathcal H$-free graph $G$ has bounded $μ(G)$ for $μ$ related to the domination number. Specifically, we consider the parameters $O\!I\!R(G)$, $I\!S(G)$, or $I\!R\!S(G)\}$, where $O\!I\!R(G)$, $I\!S(G)$, and $I\!R\!S(G)$ are the open irredundance number, the independence saturation number, and the irredundance saturation number of graph $G$, respectively.

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arXiv2024

A Ramsey-type theorem on deficiency

Ramsey's Theorem states that a graph $G$ has bounded order if and only if $G$ contains no complete graph $K_n$ or empty graph $E_n$ as its induced subgraph. The Gyárfás-Sumner conjecture says that a graph $G$ has bounded chromatic number if and only if it contains no induced subgraph isomorphic to $K_n$ or a tree $T$. The deficiency of a graph is the number of vertices that cannot be covered by a maximum matching. In this paper, we prove a Ramsey type theorem for deficiency, i.e., we characterize all the forbidden induced subgraphs for graphs $G$ with bounded deficiency. As an application, we answer a question proposed by Fujita, Kawarabayashi, Lucchesi, Ota, Plummer and Saito (JCTB, 2006).

math.CO↗

arXiv2024

Counting triangles in regular graphs

In this paper, we investigate the minimum number of triangles, denoted by $t(n,k)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andrásfai-Erdős-Sós Theorem has established that $t(n,k)>0$ if $k>\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t(n,k)$ for sufficiently large $n$, given that $\frac{2n}{5}+\frac{12\sqrt{n}}{5}<k<\frac{n}{2}$. Here, we bridge the gap between the aforementioned results by determining the precise value of $t(n,k)$ in the entire range $\frac{2n}{5}<k<\frac{n}{2}$. This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large $n$.

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arXiv2024

Long antipaths and anticycles in oriented graphs

Let $δ^{0}(D)$ be the minimum semi-degree of an oriented graph $D$. Jackson (1981) proved that every oriented graph $D$ with $δ^{0}(D)\geq k$ contains a directed path of length $2k$ when $|V(D)|>2k+2$, and a directed Hamilton cycle when $|V(D)|\le 2k+2$. Stein~(2020) further conjectured that every oriented graph $D$ with $δ^{0}(D)>k/2$ contains any orientated path of length $k$. Recently, Klimousová and Stein (DM, 2023) introduced the minimum pseudo-semi-degree $\tildeδ^0(D)$ (a slight weaker than the minimum semi-degree condition as $\tildeδ^0(D)\ge δ^0(D))$ and showed that every oriented graph $D$ with $\tildeδ^{0}(D)\ge (3k-2)/4$ contains each antipath of length $k$ for $k\geq 3$. In this paper, we improve the result of Klimousová and Stein by showing that for all $k\geq 2$, every oriented graph with $\tildeδ^0(D)\ge(2k+1)/3$ contains either an antipath of length at least $k+1$ or an anticycle of length at least $k+1$. Furthermore, we answer a problem raised by Klimousová and Stein in the negative.

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arXiv2023

A new connectivity bound for a tournament to be highly linked

A digraph $D$ is $k$-linked if for any pair of two disjoint sets $\{x_{1},x_{2},\ldots,x_{k}\}$ and $\{y_{1},y_{2},\ldots,y_{k}\}$ of vertices in $D$, there exist vertex disjoint dipaths $P_{1},P_{2},\ldots,P_{k}$ such that $P_{i}$ is a dipath from $x_{i}$ to $y_{i}$ for each $i\in[k]$. Pokrovskiy (JCTB, 2015) confirmed a conjecture of Kühn et al. (Proc. Lond. Math. Soc., 2014) by verifying that every $452k$-connected tournament is $k$-linked. Meng et al. (Eur. J. Comb., 2021) improved this upper bound by showing that any $(40k-31)$-connected tournament is $k$-linked. In this paper, we show a better upper bound by proving that every $\lceil 12.5k-6\rceil$-connected tournament with minimum out-degree at least $21k-14$ is $k$-linked. Furthermore, we improve a key lemma that was first introduced by Pokrovskiy (JCTB, 2015) and later enhanced by Meng et al. (Eur. J. Comb., 2021).

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arXiv2023

Ramsey numbers of color critical graphs versus large generalized fans

Given two graphs $G$ and $H$, the {Ramsey number} $R(G,H)$ is the smallest positive integer $N$ such that every 2-coloring of the edges of $K_{N}$ contains either a red $G$ or a blue $H$. Let $K_{N-1}\sqcup K_{1,k}$ be the graph obtained from $K_{N-1}$ by adding a new vertex $v$ connecting $k$ vertices of $K_{N-1}$. Hook and Isaak (2011) defined the {\em star-critical Ramsey number} $r_{*}(G,H)$ as the smallest integer $k$ such that every 2-coloring of the edges of $K_{N-1}\sqcup K_{1,k}$ contains either a red $G$ or a blue $H$, where $N=R(G, H)$. For sufficiently large $n$, Li and Rousseau~(1996) proved that $R(K_{k+1},K_{1}+nK_{t})=knt +1$, Hao, Lin~(2018) showed that $r_{*}(K_{k+1},K_{1}+nK_{t})=(k-1)tn+t$; Li and Liu~(2016) proved that $R(C_{2k+1}, K_{1}+nK_{t})=2nt+1$, and Li, Li, and Wang~(2020) showed that $r_{*}(C_{2m+1},K_{1}+nK_{t})=nt+t$. A graph $G$ with $χ(G)=k+1$ is called edge-critical if $G$ contains an edge $e$ such that $χ(G-e)=k$. In this paper, we extend the above results by showing that for an edge-critical graph $G$ with $χ(G)=k+1$, when $k\geq 2$, $t\geq 2$ and $n$ is sufficiently large, $R(G, K_{1}+nK_{t})=knt+1$ and $r_{*}(G,K_{1}+nK_{t})=(k-1)nt+t$.

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arXiv2023

Multicolor Ramsey numbers on stars versus pat

For given simple graphs $H_1,H_2,\dots,H_c$, the multicolor Ramsey number $R(H_1,H_2,\dots,H_c)$ is defined as the smallest positive integer $n$ such that for an arbitrary edge-decomposition $\{G_i\}^c_{i=1}$ of the complete graph $K_n$, at least one $G_i$ has a subgraph isomorphic to $H_i$. Let $m,n_1,n_2,\dots,n_c$ be positive integers and $Σ=\sum_{i=1}^{c}(n_i-1)$. Some bounds and exact values of $R(K_{1,n_1},\dots,K_{1,n_c},P_m)$ have been obtained in literature. Wang (Graphs Combin., 2020) conjectured that if $Σ\not\equiv 0\pmod{m-1}$ and $Σ+1\ge (m-3)^2$, then $R(K_{1,n_1},\ldots, K_{1,n_c}, P_m)=Σ+m-1.$ In this note, we give a new lower bound and some exact values of $R(K_{1,n_1},\dots,K_{1,n_c},P_m)$ when $m\leqΣ$, $Σ\equiv k\pmod{m-1}$, and $2\leq k \leq m-2$. These results partially confirm Wang's conjecture.

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arXiv2023

The degree threshold for covering with all the connected $3$-graphs with $3$ edges

Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an $F$-covering if every vertex in $H$ is contained in a copy of $F$. Let $c_{i}(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with $δ_{i}(H)>c_i(n,F)$ has an $F$-covering. Falgas-Ravry, Markstöm and Zhao (Combin. Probab. Comput., 2021) asymptotically determined $c_1(n,K_{4}^{(3)-})$, where $K_{4}^{(3)-}$ is obtained by deleting an edge from the complete $3$-graph on $4$ vertices. Later, Tang, Ma and Hou (arXiv, 2022) asymptotically determined $c_1(n,C_{6}^{(3)})$, where $C_{6}^{(3)}$ is the linear triangle, i.e. $C_{6}^{(3)}=([6],\{123,345,561\})$. In this paper, we determine $c_1(n,F_5)$ asymptotically, where $F_5$ is the generalized triangle, i.e. $F_5=([5],\{123,124,345\})$. We also determine the exact values of $c_1(n,F)$, where $F$ is any connected $3$-graphs with $3$ edges and $F\notin\{K_4^{(3)-}, C_{6}^{(3)}, F_5\}$.

math.CO↗

arXiv2023

Graphs without rainbow cliques of orders four and five

Let $\mathcal{G}_n^k=\{G_1,G_2,\ldots,G_k\}$ be a multiset of graphs on vertex set $[n]$ and let $F$ be a fixed graph with edge set $F=\{e_1, e_2,\ldots, e_m\}$ and $k\ge m$. We say ${\mathcal{G}_n^k}$ is rainbow $F$-free if there is no $\{i_1, i_2,\ldots, i_{m}\}\subseteq[k]$ satisfying $e_j\in G_{i_j}$ for every $j\in[m]$. Let $\ex_k(n,F)$ be the maximum $\sum_{i=1}^{k}|G_i|$ among all the rainbow $F$-free multisets ${\mathcal{G}_n^k}$. Keevash, Saks, Sudakov, and Verstraëte (2004) determined the exact value of $\ex_k(n, K_r)$ when $n$ is sufficiently large and proposed the conjecture that the results remain true when $n\ge Cr^2$ for some constant $C$. Recently, Frankl (2022) confirmed the conjecture for $r=3$ and all possible values of $n$. In this paper, we determine the exact value of $\ex_k(n, K_r)$ for $n\ge r-1$ when $r=4$ and $5$, i.e. the conjecture of Keevash, Saks, Sudakov, and Verstraëte is true for $r\in\{4,5\}$.

math.CO↗

arXiv2023

On Induced Subgraph of Cartesian Product of Paths

Chung, Füredi, Graham, and Seymour (JCTA, 1988) constructed an induced subgraph of the hypercube $Q^n$ with $α(Q^n)+1$ vertices and with maximum degree smaller than $\lceil \sqrt{n} \rceil$. Subsequently, Huang (Annals of Mathematics, 2019) proved the Sensitivity Conjecture by demonstrating that the maximum degree of such an induced subgraph of hypercube $Q^n$ is at least $\lceil \sqrt{n} \rceil$, and posed the question: Given a graph $G$, let $f(G)$ be the minimum of the maximum degree of an induced subgraph of $G$ on $α(G)+1$ vertices, what can we say about $f(G)$? In this paper, we investigate this question for Cartesian product of paths $P_m$, denoted by $P_m^k$. We determine the exact values of $f(P_{m}^k)$ when $m=2n+1$ by showing that $f(P_{2n+1}^k)=1$ for $n\geq 2$ and $f(P_3^k)=2$, and give a nontrivial lower bound of $f(P_{m}^k)$ when $m=2n$ by showing that $f(P_{2n}^k)\geq \lceil \sqrt{β_nk}\rceil$. In particular, when $n=1$, we have $f(Q^k)=f(P_{2}^k)\ge \sqrt{k}$, which is Huang's result. The lower bounds of $f(P_{3}^k)$ and $f(P_{2n}^k)$ are given by using the spectral method provided by Huang.

math.CO↗

arXiv2023

Spectral extrema of graphs with bounded clique number and matching number

For a set of graphs $\mathcal{F}$, let $\ex(n,\mathcal{F})$ and $\spex(n,\mathcal{F})$ denote the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal{F}$-free graph, respectively. Nikiforov ({\em LAA}, 2007) gave the spectral version of the Turán Theorem by showing that $\spex(n, K_{k+1})=λ(T_{k}(n))$, where $T_k(n)$ is the $k$-partite Turán graph on $n$ vertices. In the same year, Feng, Yu and Zhang ({\em LAA}) determined the exact value of $\spex(n, M_{s+1})$, where $M_{s+1}$ is a matching with $s+1$ edges. Recently, Alon and Frankl~(arXiv2210.15076) gave the exact value of $\ex(n,\{K_{k+1},M_{s+1}\})$. In this article, we give the spectral version of the result of Alon and Frankl by determining the exact value of $\spex(n,\{K_{k+1},M_{s+1}\})$ when $n$ is large.

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arXiv2022

The degree and codegree threshold for linear triangle covering in 3-graphs

Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if every vertex in $G$ is contained in a copy of $F$. For $1\le i \le k-1$, let $c_i(n,F)$ be the least integer such that every $n$-vertex $k$-uniform hypergraph $G$ with $δ_i(G)> c_i(n,F)$ has an $F$-covering. The covering problem has been systematically studied by Falgas-Ravry and Zhao [Codegree thresholds for covering 3-uniform hypergraphs, SIAM J. Discrete Math., 2016]. Last year, Falgas-Ravry, Markström, and Zhao [Triangle-degrees in graphs and tetrahedron coverings in 3-graphs, Combinatorics, Probability and Computing, 2021] asymptotically determined $c_1(n, F)$ when $F$ is the generalized triangle. In this note, we give the exact value of $c_2(n, F)$ and asymptotically determine $c_1(n, F)$ when $F$ is the linear triangle $C_6^3$, where $C_6^3$ is the 3-uniform hypergraph with vertex set $\{v_1,v_2,v_3,v_4,v_5,v_6\}$ and edge set $\{v_1v_2v_3,v_3v_4v_5,v_5v_6v_1\}$.

math.CO↗

arXiv2022

Maximum size of $C_{\leq k}$-free strong digraphs with out-degree at least two

Let $\mathscr{H}$ be a family of digraphs. A digraph $D$ is \emph{$\mathscr{H}$-free} if it contains no isomorphic copy of any member of $\mathscr{H}$. For $k\geq2$, we set $C_{\leq k}=\{C_{2}, C_{3},\ldots,C_{k}\}$, where $C_{\ell}$ is a directed cycle of length $\ell\in\{2,3,\ldots,k\}$. Let $D_{n}^{k}(ξ,ζ)$ denote the family of \emph{${C}_{\le k}$-free} strong digraphs on $n$ vertices with every vertex having out-degree at least $ξ$ and in-degree at least $ζ$, where both $ξ$ and $ζ$ are positive integers. Let $φ_{n}^{k}(ξ,ζ)=\max\{|A(D)|:\;D\in D_{n}^{k}(ξ,ζ)\}$ and $Φ_{n}^{k}(ξ,ζ)=\{D\in D_{n}^{k}(ξ,ζ): |A(D)|=φ_{n}^{k}(ξ,ζ)\}$. Bermond et al.\;(1980) verified that $φ_{n}^{k}(1,1)=\binom{n-k+2}{2}+k-2$. Chen and Chang\;(2021) showed that $\binom{n-1}{2}-2\leqφ_{n}^{3}(2,1)\leq\binom{n-1}{2}$. This upper bound was further improved to $\binom{n-1}{2}-1$ by Chen and Chang\;(DAM, 2022), furthermore, they also gave the exact values of $φ_{n}^{3}(2,1)$ for $n\in \{7,8,9\}$. In this paper, we continue to determine the exact values of $φ_{n}^{3}(2,1)$ for $n\ge 10$, i.e., $φ_{n}^{3}(2,1)=\binom{n-1}{2}-2$ for $n\geq10$.

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arXiv2022

Minimizing the number of matchings of fixed size in a $K_s$-saturated graph

For a fixed graph $F$, a graph $G$ is said to be $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$ but does contain $F$ after the addition of any new edge. Let $M_k$ be a matching consisting of $k$ edges and $S_{n,k}$ be the join graph of a complete graph $K_k$ and an empty graph $\overline{K_{n-k}}$. In this paper, we prove that for $s \geq3$ and $k\geq 2$, $S_{n,s-2}$ contains the minimum number of $M_k$ among all $n$-vertex $K_s$-saturated graphs for sufficiently large $n$, and when $k \leq s-2$, it is the unique extremal graph. In addition, we also show that $S_{n,1}$ is the unique extremal graph when $k=2$ and $s=3$.

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arXiv2022

The cycle of length four is strictly $F$-Turán-good

Given an $(r+1)$-chromatic graph $F$ and a graph $H$ that does not contain $F$ as a subgraph, we say that $H$ is strictly $F$-Turán-good if the Turán graph $T_{r}(n)$ is the unique graph containing the maximum number of copies of $H$ among all $F$-free graphs on $n$ vertices for every $n$ large enough. Győri, Pach and Simonovits (1991) proved that cycle $C_4$ of length four is strictly $K_{r+1}$-Turán-good for all $r\geq 2$. In this article, we extend this result and show that $C_4$ is strictly $F$-Turán-good, where $F$ is an $(r+1)$-chromatic graph with $r\ge 2$ and a color-critical edge. Moreover, we show that every $n$-vertex $C_4$-free graph $G$ with $N(H,G)=\ex(n,C_4,F)-o(n^4)$ can be obtained by adding or deleting $o(n^2)$ edges from $T_r(n)$. Our proof uses the flag algebra method developed by Razborov (2007).

math.CO↗

arXiv2022

An Improved Probability Propagation Algorithm for Density Peak Clustering Based on Natural Nearest Neighborhood

Clustering by fast search and find of density peaks (DPC) (Since, 2014) has been proven to be a promising clustering approach that efficiently discovers the centers of clusters by finding the density peaks. The accuracy of DPC depends on the cutoff distance ($d_c$), the cluster number ($k$) and the selection of the centers of clusters. Moreover, the final allocation strategy is sensitive and has poor fault tolerance. The shortcomings above make the algorithm sensitive to parameters and only applicable for some specific datasets. To overcome the limitations of DPC, this paper presents an improved probability propagation algorithm for density peak clustering based on the natural nearest neighborhood (DPC-PPNNN). By introducing the idea of natural nearest neighborhood and probability propagation, DPC-PPNNN realizes the nonparametric clustering process and makes the algorithm applicable for more complex datasets. In experiments on several datasets, DPC-PPNNN is shown to outperform DPC, K-means and DBSCAN.

cs.LG↗

arXiv2022

Some exact results of the generalized Turán numbers for paths

For graphs $H$ and $F$ with chromatic number $χ(F)=k$, we call $H$ strictly $F$-Turán-good (or $(H, F)$ strictly Turán-good) if the Turán graph $T_{k-1}(n)$ is the unique $F$-free graph on $n$ vertices containing the largest number of copies of $H$ when $n$ is large enough. Let $F$ be a graph with chromatic number $χ(F)\geq 3$ and a color-critical edge and let $P_\ell$ be a path with $\ell$ vertices. Gerbner and Palmer (2020, arXiv:2006.03756) showed that $(P_3, F)$ is strictly Turán good if $χ(H)\ge 4$ and they conjectured that (a) this result is true when $χ(F)=3$, and, moreover, (b) $(P_\ell, K_k)$ is Turán-good for every pair of integers $\ell$ and $k$. In the present paper, we show that $(H, F)$ is strictly Turán-good when $H$ is a bipartite graph with matching number $ν(H)=\lfloor \frac{|V(H)|}{2}\rfloor$ and $χ(F)= 3$, as a corollary, this result confirms the conjecture (a); we also prove that $(P_\ell, F)$ is strictly Turán-good for $2\le\ell\leq 6$ and $χ(F)\ge 4$, this also confirms the conjecture (b) for $2\le\ell\leq 6$ and $k\ge 4$.

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arXiv2022

A note on shortest circuit cover of 3-edge colorable cubic signed graphs

A {sign-circuit cover} $\mathcal{F}$ of a signed graph $(G, σ)$ is a family of sign-circuits which covers all edges of $(G, σ)$. The shortest sign-circuit cover problem was initiated by Má$\check{\text{c}}$ajová, Raspaud, Rollová, and Škoviera (JGT 2016) and received many attentions in recent years. In this paper, we show that every flow-admissible 3-edge colorable cubic signed graph $(G, σ)$ has a sign-circuit cover with length at most $\frac{20}{9} |E(G)|$.

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