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	1. Ramsey numbers of multiple copies of graphs in a component
	HUANG CaiXia,PENG YueJian
	Mathematics 09 May 2023
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	Abstract:For a graph $G$, $R(c(nG))$ denotes the least positive integer $N$ such that every 2-colouring of the edges of $K_N$ contains a copy of $nG$ in a monochromatic component, where $nG$ denotes the graph consisting of $n$ vertex disjoint copies of $G$.Gy\'{a}f\'{a}s and S\'{a}rk\"{o}zy showed that $R(c(nK_3))=7n-2$ for $n \geq 2$ in 2016.After that, Roberts showed that $R(c(nK_r))=(r^2-r+1)n-r+1$ for $r \geq 4$ and $n \geq R(K_r)$ in 2017.This paper determines the values of $R(c(n(K_{1,3}+e)))$ and $R(c(n(K_4-e)))$.
	TO cite this article:HUANG CaiXia,PENG YueJian. Ramsey numbers of multiple copies of graphs in a component[OL].[ 9 May 2023] http://en.paper.edu.cn/en_releasepaper/content/4760562


	2. Lattice paths and the Prouhet-Thue-Morse sequence
	Bing Gao,Shishuo Fu
	Mathematics 16 March 2023
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	Abstract:Using Flajolet's combinatorial theory of continued fractions and axial symmetry of lattice paths, we show that the enumeration of certain lattice paths modulo two yields the ubiquitous Prouhet-Thue-Morse sequence. This answers an open problem of Berstel et al.
	TO cite this article:Bing Gao,Shishuo Fu. Lattice paths and the Prouhet-Thue-Morse sequence[OL].[16 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759531

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	3. Resistance distance and the Kirchhoff index-based graph invariants of hexagonalization of graphs
	XU Can , YANG Yu-Jun
	Mathematics 15 April 2022
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	Abstract:Let $G$ be a connected graph. we consider the resistance distance and Kirchhoff index of the hexagonalization graph $H(G)$ of a graph $G$. The haxagonalization graph $H(G)$ is the graph obtained from $G$ by replacing each edge of $G$ by two disjoint paths of length 3 with the end vertices of the edge as their end vertices. Let $H^{k}(G)$ denote the the graph obtained from $G$ by $k$-iterated hexagonalizations. Using algebraic and combinatorial methods, explict formulae for resistance distances and Kirchhoff indices of $H(G)$ and $H^{k}(G)$ are obtained. It turns out that resistance distances and Kirchhoff indices of $H(G)$ and $H^{k}(G)$ could be expressed in terms of resistance distances and related structural graph invariants of $G$.
	TO cite this article:XU Can , YANG Yu-Jun. Resistance distance and the Kirchhoff index-based graph invariants of hexagonalization of graphs[OL].[15 April 2022] http://en.paper.edu.cn/en_releasepaper/content/4757250


	4. Affine Extensions of Hyperplane Arrangements
	CAI Hang,FU Hou-Shan,WANG Sui-Jie
	Mathematics 11 March 2022
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	Abstract:This paper aims to classify affine one-element extensions of an affine hyperplane arrangement. Additionally, we further establish an upper semi-continuity on the coefficients of Whitney polynomials, Whitney numbers, face numbers and region numbers among all classes.
	TO cite this article:CAI Hang,FU Hou-Shan,WANG Sui-Jie. Affine Extensions of Hyperplane Arrangements[OL].[11 March 2022] http://en.paper.edu.cn/en_releasepaper/content/4756439


	5. Two Bijections on NBC Subsets
	FU Hou-Shan,PENG Bao-Ren,WANG Sui-Jie
	Mathematics 02 March 2022
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	Abstract:In this paper, we will establish two explicit bijections: from acyclic reorientations of an oriented matroid to all no broken circuit (NBC) subsets of its underlying matroid, and from regions of an affine hyperplane arrangement to its affine NBC subsets.
	TO cite this article:FU Hou-Shan,PENG Bao-Ren,WANG Sui-Jie. Two Bijections on NBC Subsets[OL].[ 2 March 2022] http://en.paper.edu.cn/en_releasepaper/content/4756443


	6. The Lagrangian density of $\{123,234,456\}$ and the Tur\'an number of its extension
	LIANG Jinhua,PENG Yuejian
	Mathematics 27 April 2018
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	Abstract:Given a positive integer $n$ and an $r$-uniform hypergraph $F$, the {\em Tur\'an number} $ex(n,F)$ is the maximum number of edges in an $F$-free $r$-uniform hypergraph on $n$ vertices. The {\em Tur\'{a}n density} of $F$ is defined as $\pi(F)=\lim_{n\rightarrow\infty} { ex(n,F) \over {n \choose r } }.$ The {\em Lagrangian density } of $F$ is$\pi_{\lambda}(F)=\sup \{r! \lambda(G):G\;is\;F\text{-}free\}.$ Sidorenko observed that $\pi(F)\le \pi_{\lambda}(F)$, and Pikhurko observed that $\pi(F)=\pi_{\lambda}(F)$ if every pair of vertices in $F$ is contained in an edge of $F$. Recently, Lagrangian densities of hypergraphs and Tur\'{a}n numbers of their extensions have been studied actively. For example, in the paper `A hypergraph Tur\'an theorem via Lagrangians of intersectingfamilies, {\it J. Combin. Theory Ser. A} {\bf 120} (2013), 2020--2038.', Hefetz and Keevash studied the Lagrangian densitiy of the $3$-uniform graph spanned by $\{123, 456\}$ and the Tur\'{a}n number of its extension.In this paper, we show that the Lagrangian density of the $3$-uniform graph spanned by $\{123,234,456\}$ achieves only on $K_5^3$ . Applying it, we get the Tur\'{a}n number of its extension, and show that the unique extremal hypergraph is the balanced complete $5$-partite $3$-uniform hypergraph on $n$ vertices.
	TO cite this article:LIANG Jinhua,PENG Yuejian. The Lagrangian density of $\{123,234,456\}$ and the Tur\'an number of its extension[OL].[27 April 2018] http://en.paper.edu.cn/en_releasepaper/content/4744794


	7. Degree Sequence for $k$-arc Strong Connected Multiple Digraphs
	Qinghai Liu,Yanmei Hong
	Mathematics 12 May 2017
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	Abstract:Let $D$ be a digraph on ${v_1,ldots, v_n}$. Then the sequence ${(d^+(v_1), d^-(v_1)), ldots,(d^+(v_n), d^-(v_n))}$ is called the degree sequence of $D$. For any given a sequence of pairs of integers $mathbf{d}={(d_1^+, d_1^-), ldots, (d_n^+, d_n^-)}$, if there exists a $k$-arc strong connected digraph $D$ such that$mathbf d$ is the degree sequence of $D$ then $mathbf d$ is realizable and $D$ is a realization of $mathbf d$. In this paper, a characterization for a $k$-arc connected realizable sequence is given and it can be checked in a linear time.
	TO cite this article:Qinghai Liu,Yanmei Hong. Degree Sequence for $k$-arc Strong Connected Multiple Digraphs[OL].[12 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4731143


	8. Circumference of 3-connected cubic graphs
	Qinghai Liu,Xingxing Yu,Zhao Zhang
	Mathematics 12 May 2017
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	Abstract:The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $Omega(n^{0.694})$.Bilinski {it et al.} improved this lower bound to $Omega(n^{0.753})$ bystudying large Eulerian subgraphs in 3-edge-connected graphs.In this paper, we further improve this lower bound to $Omega(n^{0.8})$.This is done by considering certain 2-connected cubic graphs, finding cycles through two given edges, anddistinguishing the cases whether or not these edges are adjacent.
	TO cite this article:Qinghai Liu,Xingxing Yu,Zhao Zhang. Circumference of 3-connected cubic graphs[OL].[12 May 2017] http://en.paper.edu.cn/en_releasepaper/content/4731498


	9. On sequences without short zero-sum subsequences
	Zeng Xiangneng,Yuan Pingzhi
	Mathematics 31 March 2017
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	Abstract:Let $G$ be a finite abelian group. It is well known that every sequence $S$ over $G$ of length at least $\|G\|$ contains a zero-sum subsequence of length at most $mathsf{h}(S)$. It is interesting to study the corresponding inverse problem, that is to find the information on the structure of the sequence $S$ which does not contain zero-sum subsequences of length at most $mathsf{h}(S)$. Under the assumption that $\|sum(S)\|< min{\|G\|,2\|S\|-1}$, Gao, Peng and Wang showed that such a sequence $S$ must be strictly behaving. In the present paper, we take a step forward and give explicitly the structure of such a sequence $S$ under the assumptionthat $\|sum(S)\|=2\|S\|-1<\|G\|$.
	TO cite this article:Zeng Xiangneng,Yuan Pingzhi. On sequences without short zero-sum subsequences[OL].[31 March 2017] http://en.paper.edu.cn/en_releasepaper/content/4723585


	10. Inversion arrangements and the weak Bruhat order
	Neil J.Y. Fan
	Mathematics 22 January 2017
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	Abstract:For each permutation $w$, we can construct a collection of hyperplanes $mathcal{A}_w$ according to the inversions of $w$, which is called the inversion hyperplane arrangement associated to $w$. It was conjectured by Postnikov and confirmed by Hultman et al. that the number of regions of $mathcal{A}_w$ is less than or equal to the number of permutations below $w$ in the Bruhat order, with the equality holds if and only if $w$ avoids the four patterns 4231, 35142, 42513 and 351624. In this paper, we show that the number of regions of $mathcal{A}_w$ is greater than or equal to the number of permutations below $w$ in the weak Bruhat order, with the equality holds if and only if $w$ avoids the patterns 231 and 312.
	TO cite this article:Neil J.Y. Fan. Inversion arrangements and the weak Bruhat order[OL].[22 January 2017] http://en.paper.edu.cn/en_releasepaper/content/4717491

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