含偶长圈的7点7边图的图设计

设λKν是ν阶λ重完全图，G是一个无孤立点的有限简单图,λKν的一个G-分拆(或G-设计，记为G-GDλ(ν))是指一个序偶(X，β)，其中X是完全图Kν的顶点集，β是Kν中同构于G的子图(称为区组)的族，使得Kν中每条边恰好出现在β的λ个区组中，本文完全解决了含偶长圈的十个7点7边图的图设计存在性问题。     

On K_(1,k)-factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n,which partition the set of edges of λKm,n.In this paper,it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n,whenever k is any positive integer,is that(1) m ≤ kn,(2) n ≤ km,(3) km-n ≡ kn-m ≡ 0(mod(k2-1)) and(4) λ(km-n)(kn-m) ≡ 0(mod k(k -1)(k2 -1)(m n)).     

平衡子欧拉符号图及符号线图

符号图$S=(S^u,\sigma)$是以$S^u$作为底图并且满足$\sigma: E(S^u)\rightarrow\{+,-\}$. 设$E^-(S)$表示$S$的负边集. 如果$S^u$是欧拉的(或者分别是子欧拉的, 欧拉的且$|E^-(S)|$是偶数, 则$S$是欧拉符号图(或者分别是子欧拉符号图, 平衡欧拉符号图). 如果存在平衡欧拉符号图$S''$使得$S''$由$S$生成, 则$S$是平衡子欧拉符号图. 符号图$S$的线图$L(S)$也是一个符号图, 使得$L(S)$的点是$S$中的边, 其中$e_ie_j$是$L(S)$中的边当且仅当$e_i$和$e_j$在$S$中相邻,并且$e_ie_j$是$L(S)$中的负边当且仅当$e_i$和$e_j$在$S$中都是负边. 本文给出了两个符号图族$S$和$S''$,它们应用于刻画平衡子欧拉符号图和平衡子欧拉符号线图. 特别地, 本文证明了符号图$S$是平衡子欧拉的当且仅当$\not\in S$, $S$的符号线图是平衡子欧拉的当且仅当$S\not\in S''$.     

Decomposing Complete Equipartite Multigraphs into Cycles of Variable Lengths: The Amalgamation‐detachment Approach

Using the technique of amalgamation‐detachment, we show that the complete equipartite multigraph can be decomposed into cycles of lengths (plus a 1‐factor if the degree is odd) whenever there exists a decomposition of into cycles of lengths (plus a 1‐factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph .     

Analyzing local and global properties of multigraphs

The local structure of undirected multigraphs under two random multigraph models is analyzed and compared. The first model generates multigraphs by randomly coupling pairs of stubs according to a fixed degree sequence so that edge assignments to vertex pair sites are dependent. The second model is a simplification that ignores the dependency between the edge assignments. It is investigated when this ignorance is justified so that the simplified model can be used as an approximation, thus facilitating the structural analysis of network data with multiple relations and loops. The comparison is based on the local properties of multigraphs given by marginal distribution of edge multiplicities and some local properties that are aggregations of global properties.     

Classical Kloosterman sums: Representation theory, magic squares, and Ramanujan multigraphs

We consider a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of “magical” combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.     

Complexity of the pipeline computation of a family of inner products

We are interested in the minimum time T(S) necessary for computing a family S = { < S_i, S_j >: ? S_i, S_j?R^p, (i, j) ?E } of inner products of order p, on a systolic array of order p × 2. We first prove that the determination of T(S) is equivalent to the partition problem and is thus NP-complete. Then we show that the designing of an algorithm which runs in time T(S) + 1 is equivalent to the problem of finding an undirected bipartite eulerian multigraph with the smallest number of edges, which contains a given undirected bipartite graph, and can therefore be solved in polynomial time.     

Supereulerian Digraphs with Large Arc‐Strong Connectivity

Let D be a digraph and let be the arc‐strong connectivity of D, and be the size of a maximum matching of D. We proved that if , then D has a spanning eulerian subdigraph.     

关于重图的边色数

本文利用 A.Ehrenfeucht[2 ]等提出的两个引理 ,给出两个重图边色数的定理 .     

Star number and star arboricity of a complete multigraph

Let G be a multigraph. The star number s(G) of G is the minimum number of stars needed to decompose the edges of G. The star arboricity sa(G) of G is the minimum number of star forests needed to decompose the edges of G. As usual λK _n denote the λ-fold complete graph on n vertices (i.e., the multigraph on n vertices such that there are λ edges between every pair of vertices). In this paper, we prove that for n ⩾ 2