The Non-Collinearity Graph of the O+(8,2) Quadric Is Uniquely Geometrisable

The graph of the titlehas the points of the O⁺(8,2) polar space as itsvertices, two such vertices being adjacent iff the correspondingpoints are non-collinear in the polar space. We prove that, uptoisomorphism, there is a unique partial geometry pg(8,7,4)whose point graph is this graph. This is the partial geometryof Cohen, Haemers and Van Lint and De Clerck, Dye and Thas. Ouruniqueness proof shows that this geometry has a subgeometry isomorphicto the affine plane of order three, and the geometry is canonicallydescribeable in terms of this affine plane.     

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m,n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that n/2 -2 g(4,n) (n/2) + o(n) and n/4 g(n,4) (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4,n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that n/4 p(4,n) (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m,n).     

The color space of a graph

If k is a prime power, and G is a graph with n vertices, then a k‐coloring of G may be considered as a vector in $\mathbb{GF}$(k)ⁿ. We prove that the subspace of $\mathbb{GF}$(3)ⁿ spanned by all 3‐colorings of a planar triangle‐free graph with n vertices has dimension n. In particular, any such graph has at least n − 1 nonequivalent 3‐colorings, and the addition of any edge or any vertex of degree 3 results in a 3‐colorable graph. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 234–245, 2000     

A Polynomial-Time Algorithm for Finding Regular Simple Paths in Outerplanar Graphs

Let G be a labeled directed graph with arc labels drawn from alphabet Σ, R be a regular expression over Σ, and x and y be a pair of nodes from G. The regular simple path (RSP) problem is to determine whether there is a simple path p in G from x to y, such that the concatenation of arc labels along p satisfies R. Although RSP is known to be NP-hard in general, we show that it is solvable in polynomial time when G is outerplanar. The proof proceeds by presenting an algorithm which gives a polynomial-time reduction of RSP for outerplanar graphs to RSP for directed acyclic graphs, a problem which has been shown to be solvable in polynomial time.     

Banach空间一类Lipschitz映射不动点的迭代逼近

文章利用正规对偶映射的定义,给出了任意Banach空间Lipschitz强伪压缩映射不动点的Ishikawa迭代收敛定理．该定理不仅推广了已知结果,而且还简化了目前相应结果的证明．     

Between clique-width and linear clique-width of bipartite graphs

We consider hereditary classes of bipartite graphs where clique-width is bounded, but linear clique-width is not. Our goal is identifying classes that are critical with respect to linear clique-width. We discover four such classes and conjecture that this list is complete, i.e. a hereditary class of bipartite graphs of bounded clique-width that excludes a graph from each of the four critical classes has bounded linear clique-width.     

Vertex-disjoint rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph of order $n$. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if ${\delta }^c\left(G\right)\ge (n+1)∕2$, then $G$ contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if $n\ge 20$ and ${\delta }^c\left(G\right)\ge (n+2)∕2$, then $G$ contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if ${\delta }^c\left(G\right)\ge (n+k)∕2$, then $G$ contains $k$ vertex-disjoint rainbow triangles. For any integer $k\ge 2$, we show that if $n\ge 16k-12$ and ${\delta }^c\left(G\right)\ge n∕2+k-1$, then $G$ contains $k$ vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of $k$ edge-disjoint rainbow triangles.     

On the number of rainbow spanning trees in edge-colored complete graphs

A spanning tree of a properly edge-colored complete graph, $K_n$, is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if $K_{2m}$ is properly $(2m?1)$-edge-colored, then the edges of $K_{2m}$ can be partitioned into $m$ rainbow spanning trees except when $m=2$. By means of an explicit, constructive approach, in this paper we construct $?\sqrt{6m+9}∕3?$ mutually edge-disjoint rainbow spanning trees for any positive value of $m$. Not only are the rainbow trees produced, but also some structure of each rainbow spanning tree is determined in the process. This improves upon best constructive result to date in the literature which produces exactly three rainbow trees.     

极大加可约矩阵的谱与周期时间向量的关系

提出了极大加代数上可约矩阵特征值的缺失值及冗余值的概念,得到了相应的定理;对特征值与周期时间向量分量之间的关系作了深入的研究.