Base-cover paracompactness

Call a topological space base-cover paracompact if has an open base such that every cover of contains a locally finite subcover. A subspace of the Sorgenfrey line is base-cover paracompact if and only if it is . The countable sequential fan is not base-cover paracompact. A paracompact space is locally compact if and only if its product with every compact space is base-cover paracompact.

     

A note on unimodular congruence of the Laplacian matrix of a graph

Let G be a simple connected graph and L(G) be its Laplacian matrix. In this note, we prove that L(G) is congruent by a unimodular matrix to its Smith normal form if and only if G is a tree.     

关于扇和完全等二部图联图的点可区别边染色

通过结构分析的方法,考虑各种不同情况,给出了一类联图的点可区别的边染色方法,并得到了它的点可区别的边色数.     

On fans in multigraphs

We introduce a unifying framework for studying edge‐coloring problems on multigraphs. This is defined in terms of a rooted directed multigraph , which is naturally associated to the set of fans based at a given vertex u in a multigraph G. We call the “Fan Digraph.” We show that fans in G based at u are in one‐to‐one correspondence with directed trails in starting at the root of . We state and prove a central theorem about the fan digraph, which embodies many edge‐coloring results and expresses them at a higher level of abstraction. Using this result, we derive short proofs of classical theorems. We conclude with a new, generalized version of Vizing's Adjacency Lemma for multigraphs, which is stronger than all those known to the author. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 301–318, 2006     

A new infinite class of S(3, {4, 5, 7}, v)

A t‐wise balanced design ( at BD) of order v and block sizes from K , denoted by S ( t , K , v ), is a pair ( X , ??), where X is a v ‐element set and ?? is a set of subsets of X , called blocks , with the property that | B |∈ K for any B ∈?? and every t ‐element subset of X is contained in a unique block. In this article, we shall show that there is an S ( 3 , { 4 , 5 , 7 }, v ) for any positive integer v ≡ 7 ( mod12 ) with v ≠ 19 . Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:68–80, 2012     

Toric Rigid Spaces

This paper gives a method to construct rigid spaces, which is similar to the method used to construct toric schemes.     

Restricted unimodular chordal graphs

A chordal graph is called restricted unimodular if each cycle of its vertex‐clique incidence bipartite graph has length divisible by 4. We characterize these graphs within all chordal graphs by forbidden induced subgraphs, by minimal relative separators, and in other ways. We show how to construct them by starting from block graphs and multiplying vertices subject to a certain restriction, which leads to a linear‐time recognition algorithm. We show how they are related to other classes such as distance‐hereditary chordal graphs and strongly chordal graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 121–136, 1999     

On Unimodular Rows over Polynomial Rings

Let A be a commutative ring and n 3 a positive integer. In this paper, we consider unimodular rows (f₁(x),..., f_n(x)) over A[x]. We prove that, if the row of the leading coefficients of f_i(x) is unimodular over A and a A, then there exists E_n(A[x]) such that (f₁(x),...,f_n(x)) = (f₁(a),...,f_n(a)). Also, if A is a Noetherian ring with finite Krull dimension and the row of leading coefficients satisfies the same condition, then we give a bound for the length of in terms of elementary transvections.Partially supported by INTAS 93–436EXT and DFG–RFBR grant No. 96–01–00092G.2000 Mathematics Subject Classification: 19A13, 19B14, 13C10, 13B25, 13F20     

Random graphs containing few disjoint excluded minors

The Erd?s‐Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a ‘blocking’ set B of at most f(k) vertices such that the graph G – B is acyclic. Robertson and Seymour (1986) give an extension concerning any minor‐closed class of graphs, as long as does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for , there is a set B of at most g(k) vertices such that G – B is in . In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763–775), we showed that, amongst all graphs on vertex set which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices. In the present paper we build on the previous work, and give an extension concerning any minor‐closed graph class with 2‐connected excluded minors, as long as does not contain all fans (here a ‘fan’ is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on which contain at most k disjoint excluded minors for , all but an exponentially small proportion contain a set B of k vertices such that G – B is in . (This is not the case when contains all fans.) For a random graph R_n sampled uniformly from the graphs on with at most k disjoint excluded minors for , we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 240‐268, 2014     

Pure tetrahedral quadruple systems with index two

An oriented tetrahedron defined on four vertices is a set of four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order $n$ with index $\lambda$, denoted by ${\text{TQS}}_{\lambda }\left(n\right)$, is a pair $\left(X,\mathrm{?}\right)$, where $X$ is an $n$‐set and $\mathrm{?}$ is a set of oriented tetrahedra (blocks) such that every cyclic triple on $X$ is contained in exactly $\lambda$ members of $\mathrm{?}$. A ${\text{TQS}}_{\lambda }\left(n\right)$ is pure if there do not exist two blocks with the same vertex set. When $\lambda =1$, the spectrum of a pure TQS $\left(n\right)$ has been completely determined by Ji. In this paper, we show that there exists a pure ${\text{TQS}}_2\left(n\right)$ if and only if $n\equiv 1,2\left(\mathrm{mod}3\right)$ and $n\ge 7$. A corollary is that a simple ${\text{QS}}_4\left(n\right)$ also exists if and only if $n\equiv 1,2\left(\mathrm{mod}3\right)$ and $n\ge 7$.