围长为r的n阶本原有向图的点指数

研究本原有向图的顶点指数,运用图论与数论方法,得到了n阶围长为r的本原有向图的点指数expD（k）的上界：若rn,且r为素数,D∈Dn,r={D｜D为n阶本原有向图且围长为r},则expD（n,k）=rn-2r＋k（1≤k≤n）;若r｜n,且r为素数或素数的幂,D∈Dn,r,则expD（n,1）=rn-3r＋2.     

Orientation reversing involutions and the first betti number for finite coverings of 3-manifolds

LetMS ³,P ³ be a closed, orientable irreducible 3-manifold which admits an orientation reversing involution :MM. If dim(Fix )=0, suppose ₁ (M) has a subgroup of even index. We show thatM has finite coverMMM} with ₁(M<0). As an application we show that the hyperbolic dodecahedral space has a finite cover with positive 1st betti number.     

The binding number of a graph and its Anderson number

The binding number of a graph G, bind(G), is defined; some examples of its calculation are given, and some upper bounds for it are proved. It is then proved that, if bind(G) ≥ c, then G contains at least $\text{|G|}\text{c}\text{(c + 1)}$ disjoint edges if $\text{0 ≤ c ≤}\text{1}\text{2}$, at least $\text{| G |}\text{(3c ? 2)}\text{3c}\text{?}\text{2(c ? 1)}\text{c}$ disjoint edges if $\text{1 ≤ c ≤}\text{4}\text{3}$, a Hamiltonian circuit if $\text{c ≥}\text{3}\text{2}$, and a circuit of length at least $\text{3(| G | ?1)(c ? 1)}\text{c}$ if $\text{1 < c ≤}\text{3}\text{2}$ and G is not one of two specified exceptional graphs. Each of these results is best possible.The Anderson number of a graph is defined. The Anderson numbers of a few very simple graphs are determined; and some rather weak bounds are obtained, and some conjectures made, on the Anderson numbers of graphs in general.     

The basis number of a graph

MacLane proved that a graph is planar if and only if it has a 2-fold basis for its cycle space. We define the basis number of a graph G to be the least integer k such that G has a k-fold basis for its cycle space. We investigate the basis number of the complete graphs, complete bipartite graphs, and the n-cube.     

The representability number of a chain

For each pair of linear orderings (L,M), the representability number repr_M(L) of L in M is the least ordinal α such that L can be order-embedded into the lexicographic power . The case is relevant to utility theory. The main results in this paper are as follows. (i) If κ is a regular cardinal that is not order-embeddable in M, then repr_M(κ)=κ; as a consequence, for each κω₁. (ii) If M is an uncountable linear ordering with the property that A×_lex2 is not order-embeddable in M for each uncountable AM, then for any ordinal α; in particular, . (iii) If L is either an Aronszajn line or a Souslin line, then .     

The dichromatic number of a digraph

In this paper the concept of dichromatic number of a digraph which is a generalization of the chromatic number of a graph is introduced. The dichromatic number of a digraph D is defined as the minimum number of colours required to colour the vertices of D in such a way that the chromatic classes induce acyclic subdigraphs in D. Some results relating the dichromatic number of D with the existence of cycles of special lengths in D are presented. Contributions to chromatic theory are also obtained. In particular, we generalize the theorem due to P. Erdös and A. Hajnal (Acta Math. Acad. Sci. Hungar.17 (1966), 61–99) which states the existence of odd cycles of length ≥_χ(G) ? 1 in any graph G.     

The Hoffman number of a graph

For a connected graph G with n vertices, let {λ₁,λ₂,…,λ_r} be the set of distinct positive eigenvalues of the Laplacian matrix of G. The Hoffman number μ(G) of G is defined by μ(G)=λ₁λ₂…λ_r/n. In this paper, we study some properties and applications of the Hoffman number.     

The eavesdropping number of a graph

Let G be a connected, undirected graph without loops and without multiple edges. For a pair of distinct vertices u and v, a minimum {u, v}-separating set is a smallest set of edges in G whose removal disconnects u and v. The edge connectivity of G, denoted λ(G), is defined to be the minimum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. We introduce and investigate the eavesdropping number, denoted ε(G), which is defined to be the maximum cardinality of a minimum {u, v}-separating set as u and v range over all pairs of distinct vertices in G. Results are presented for regular graphs and maximally locally connected graphs, as well as for a number of common families of graphs.     

The disconnection number of a graph

The disconnection number d(X) is the least number of points in a connected topological graph X such that removal of d(X) points will disconnect X (Nadler, 1993 [6]). Let Dn denote the set of all homeomorphism classes of topological graphs with disconnection number n. The main result characterizes the members of Dn+1 in terms of four possible operations on members of Dn. In addition, if X and Y are topological graphs and X is a subspace of Y with no endpoints, then d(X)?d(Y) and Y obtains from X with exactly d(Y)−d(X) operations. Some upper and lower bounds on the size of Dn are discussed.The algorithm of the main result has been implemented to construct the classes Dn for n?8, to estimate the size of D₉, and to obtain information on certain subclasses such as non-planar graphs (n?9) and regular graphs (n?10).     

The semi-chromatic number of a graph

We introduce the notion of the semi-chromatic number of a graph with a nonempty number of edges. Then we prove that the difference between the semi-chromatic number and the half of the chromatic number is at most 1.     

Consequences of reversing preferences

Other than standard election disruptions involving shenanigans, strategic voting, and so forth, it is reasonable to expect that elections are free from difficulties. But this is far from being true; even sincere election outcomes admit all sorts of counterintuitive conclusions.     

The star chromatic number of a graph

We study a generalization of the notion of the chromatic number of a graph in which the colors assigned to adjacent vertices are required to be, in a certain sense, far apart. © 1993 John Wiley & Sons, Inc.     

The upper traceable number of a graph

For a nontrivial connected graph G of order n and a linear ordering s: v ₁, v ₂, …, v _n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t ⁺(G) of G is t ⁺(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t ⁺(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.     

The circular chromatic number of a digraph

We introduce the circular chromatic number χ_c of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k‐cycle has circular chromatic number k/(k – 1), for k ≥ 2, values of χ_c between 1 and 2 are possible. We show that in fact, χ_c takes on all rational values greater than 1. Furthermore, there exist digraphs of arbitrarily large digirth and circular chromatic number. It is NP‐complete to decide if a given digraph has χ_c at most 2. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 227–240, 2004     

The Lawson number of a semitopological semilattice

Semigroup Forum - For a Hausdorff topologized semilattice X its Lawson number $$\bar{\Lambda }(X)$$ is the smallest cardinal $$\kappa $$ such that for any distinct points $$x,y\in X$$ there exists...     

The book crossing number of a graph

Let G be a graph on n vertices and m edges. The book crossing number of G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, such that each edge is contained by one page. Our main results are two polynomial time algorithms to generate near optimal drawing of G on books. The first algorithm give an O(log² n) times optimal solution, on small number of pages, under some restrictions. This algorithm also gives rise to the first polynomial time algorithm for approximating the rectilinear crossing number so that the coordinates of vertices in the plane are small integers, thus resolving a recent open question concerning the rectilinear crossing number. Moreover, using this algorithm we improve the best known upper bounds on the rectilinear crossing number. The second algorithm generates a drawing of G with O(m²/k²) crossings on k pages. This is within a constant multiplicative factor from our general lower bound of Ω(m³/n²k²), provided that m = Ψ(n²). © 1996 John Wiley & Sons, Inc.