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.     

The toroidal thickness of the symmetric quadripartite graph

The toroidal thickness t₁(G) of a graph G is the minimum value of k for which G is the edge-union of k graphs each embeddable on a torus. It is shown that $\text{t}{}_1\text{(K}{}_{\mathrm{4(n)}}\text{) = [}\text{1}\text{2}\text{(n + 1)]}$.     

The splittance of a graph

The splittance of an arbitrary graph is the minimum number of edges to be added or removed in order to produce a split graph (i.e. a graph whose vertex set can be partitioned into a clique and an independent set). The splittance is seen to depend only on the degree sequence of the graph, and an explicit formula for it is derived. This result allows to give a simple characterization of the degree sequences of split graphs. Worst cases for the splittance are determined for some classes of graphs (the class of all graphs, of all trees and of all planar graphs).     

The graph of a foliation

Let M be a riemannian manifold with a riemannian foliation F. Among other things we construct a special metric on the graph of the foliation, , (which is complete, when M is complete), and use the relations of Gray [1] and O'Neill [7] and the elementary structural properties of , to find a necessary and sufficient condition that also have non-positive sectional curvature, when M does. This condition depends only on the second fundamental form and the holonomy of the leaves. As a corollary we obtain a generalization of the Cartan-Hadamard Theorem. Partially supported by NSF Grant MCS77-02721.     

The point-arboricity of a graph

The point-arboricity ρ(G) of a graphG is defined as the minimum number of subsets in a partition of the point set ofG so that each subset induces an acyclic subgraph. Dually, the tuleity τ(G) is the maximum number of disjoint, point-induced, non-acyclic subgraphs contained inG. Several results concerning these numbers are presented, among which are formulas for the point arboricity and tulgeity of the class of completen-partite graphs. Definitions not given in this article may be found in [5].     

The dimension of a graph

For each graph G the dimension of G is defined as the smallest dimension in the Euclidean Space where there is an embedding in which all the edges of G are segments of a straight line of length one. The exact value is calculated for some important families of graphs and this value is compared with other invariants. An infinite quantity of forbidden graphs for dimension 2 is also shown.     

The biparticity of a graph

The biparticity β(G) of a graph G is the minimum number of bipartite graphs required to cover G. It is proved that for any graph G, β(G) = {log₂χ(G)}. In view of the recent announcement of the Four Color Theorem, it follows that the biparticity of every planar graph is 2.     

The spectral radii of a graph and its line graph

The spectral radius of a (directed) graph is the largest eigenvalue of adjacency matrix of the (directed) graph. We give the relation on the characteristic polynomials of a directed graph and its line graph, and obtain sharp bounds on the spectral radius of directed graphs. We also give the relation on the spectral radii of a graph and its line graph. As a consequence, the spectral radius of a connected graph does not exceed that of its line graph except that the graph is a path.     

The total graph and regular graph of a commutative ring

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z(R) is not an ideal. In this paper we show that if T(Γ(R)) is a connected graph, then . Also, we prove that if R is a finite ring, then T(Γ(R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.     

The entire graph of a bridgeless connected plane graph is hamiltonian

In this paper we show that the entire graph of a bridgeless connected plane graph is hamiltonian, and that the entire graph of a plane block is hamiltonian connected and vertex pancyclic. In addition, we show that in any block G which is not a circuit, given a vertex v of G and a circuit k of G, there is a path p, suspended in G, such that p is a path in k of length at least 1 and G ? E(p) ? V₀(G ? E(p)) is a block which includes v.     

The asymptotic probability that a random graph is a unit interval graph,indifference graph,or proper interval graph

Let G be a graph on v labelled vertices with E edges, without loops or multiple edges. Let v → ∞ and let E=E(v) be a function of v such that lim $\text{E(v)}\text{v}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{=c}$. The limit of the probability that a random graph is a unit interval graph, indifference graph or proper interval graph is exp(?$\text{4}\text{3}$c³).     

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.     

The edge L-function of a graph

We extend Watanabe and Fukumizu’s Theorem on the edge zeta function to a regular covering of a graph G. Next, we define an edge L-function of a graph G, and give a determinant expression of it. As a corollary, we present a decomposition formula for the edge zeta function of a regular covering of G by a product of edge L-functions of G.     

The deficiency of a regular graph

A proper edge coloring c:E(G)→Z of a finite simple graph G is an interval coloring if the colors used at each vertex form a consecutive interval of integers. Many graphs do not have interval colorings, and the deficiency of a graph is an invariant that measures how close a graph comes to having an interval coloring. In this paper we search for tight upper bounds on the deficiencies of k-regular graphs in terms of the number of vertices. We find exact values for 1?k?4 and bounds for larger k.     

The threshold weight of a graph

The threshold weight of a graph G is introduced as a measure of the amount by which G differs from being a threshold graph. The threshold graphs are precisely the graphs whose threshold weights are 0. At the opposite extreme is the class of graphs for which the threshold weight is the maximum possible. Such graphs are defined as heavy graphs. Among the results are as following: A theorem that specifies the threshold weight of any triangle-free graph; necessary and sufficient conditions for a heavy graph in terms of the solvability of a system of linear inequalities; some sufficient conditions for a graph to be heavy and a necessary condition (conjectured to be sufficient, as well) for a heavy graph in terms of its cliques.     

The integrity of a cubic graph

Integrity, a measure of network reliability, is defined as

where G is a graph with vertex set V and m(G−S) denotes the order of the largest component of G−S. We prove an upper bound of the following form on the integrity of any cubic graph with n vertices:

Moreover, there exist an infinite family of connected cubic graphs whose integrity satisfies a linear lower bound I(G)>βn for some constant β. We provide a value for β, but it is likely not best possible. To prove the upper bound we first solve the following extremal problem. What is the least number of vertices in a cubic graph whose removal results in an acyclic graph? The solution (with a few minor exceptions) is that n/3 vertices suffice and this is best possible.     

The circular dimension of a graph

A graph is a pair (V, I), V being the vertices and I being the relation of adjacency on V. Given a graph G, then a collection of functions {f_i}^m_n=1, each f_i mapping each vertex of V into anarc on a fixed circle, is said to define an m-arc intersection model for G if for all x,y ? V, xly ? (∨_i?m)(f_i(x)∩f_i(y)≠Ø). The circular dimension of a graph G is defined as the smallest integer m such that G has an m-arc intersection model. In this paper we establish that the maximum circular dimension of any complete partite graph having n vertices is the largest integer p such that 2^p+p?n+1.