A lower bound for the interval number of a graph

The interval number i(G) of a graph G with n vertices is the lowest integer m such that G is the intersection graph of some family of sets I₁,…,I_n with every I_i being the union of at most m real intervals. In this article a lower bound for i(G) is proved followed by some considerations about the construction of graphs that are critical with respect to the interval number.     

The lower bound on independence number

Let G be a graph with degree sequence ( d_v). If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least , where f_m+1( x) is a function greater than for x> 0. For a weighted graph G = ( V, E, w), we prove that its weighted independence number (the maximum sum of the weights of an independent set in G) is at least where w_v is the weight of v.     

A lower bound on the independence number of arbitrary hypergraphs

We present a lower bound on the independence number of arbitrary hypergraphs in terms of the degree vectors. The degree vector of a vertex v is given by d(v) = (d₁(v), d₂(v), …) where d_m(v) is the number of edges of size m containing v. We define a function f with the property that any hypergraph H = (V, E) satisfies α(H) ≥ Σ_v∈V f(d(v)). This lower bound is sharp when H is a match, and it generalizes known bounds of Caro/Wei and Caro/Tuza for ordinary graphs and uniform hypergraphs. Furthermore, an algorithm for computing independent sets of size as guaranteed by the lower bound is given. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 213–221, 1999     

On a lower bound for the connectivity of the independence complex of a graph

Aharoni, Berger and Ziv proposed a function which is a lower bound for the connectivity of the independence complex of a graph. They conjectured that this bound is optimal for every graph. We give two different arguments which show that the conjecture is false.     

A lower bound for the circumference of a graph

Let G=(V, E) be a block of order n, different from K_n. Let m=min {d(x)+d(y): [x, y]?E}. We show that if m?n then G contains a cycle of length at least m.     

An optimal lower bound on the number of variables for graph identification

In this paper we show that (n) variables are needed for first-order logic with counting to identify graphs onn vertices. Thek-variable language with counting is equivalent to the (k–1)-dimensional Weisfeiler-Lehman method. We thus settle a long-standing open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant becausen variables obviously suffice to identify graphs onn vertices.Research supported by NSF grant CCR-8709818.Research supported by NSF grant CCR-8805978 and Pennsylvania State University Research Initiation grant 428-45.Research supported by NSF grants DCR-8603346 and CCR-8806308.     

A bound on the chromatic number of a graph

We give an upper bound on the chromatic number of a graph in terms of its maximum degree and the size of the largest complete subgraph. Our results extends a theorem due to Brooks.     

The travelling preacher, projection, and a lower bound for the stability number of a graph

The coflow min–max equality is given a travelling preacher interpretation, and is applied to give a lower bound on the maximum size of a set of vertices, no two of which are joined by an edge.     

An upper bound for the k-domination number of a graph

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k, then γk(G) ≤ kp/(k + 1).     

An upper bound for the path number of a graph

The path number of a graph G, denoted p(G), is the minimum number of edge-disjoint paths covering the edges of G. Lovász has proved that if G has u odd vertices and g even vertices, then p(G) ≤ 1/2 u + g - 1 ≤ n - 1, where n is the total number of vertices of G. This paper clears up an error in Lovász's proof of the above result and uses an extension of his construction to show that p(G) ≤ 1/2 u + [3/4g] ≤ [3/4n].     

A lower regulator bound for number fields

Lower bounds for regulators of algebraic number fields are very important for a variety of applications. Good estimates depend at least on the degree and the discriminant of the considered field. In this paper we present an improved bound which is obtained from more specific field data, e.g. the size of small T₂-values of the integers of the field. This is of considerable interest for computations in practice, for example, of fundamental units.     

A lower bound for the number of intermediary rings

We give a sharp lower bound for the number of intermediary rings in normal pairs. As a consequence, the exact length of maximal chains, and a sharp lower bound for the number of overrings of Prüfer domains are obtained.     

A lower bound for on-line ranking number of a path

A k-ranking of a graph G is a mapping $\varphi :V(G)\to \{1,\dots ,k\}$ such that any path with endvertices x and y satisfying $x\ne y$ and $\varphi (x)=\varphi (y)$ contains a vertex z with $\varphi (z)>\varphi (x)$. The ranking number ${\chi }_{\mathrm{r}}(G)$ of G is the minimum k admitting a k-ranking of G. The on-line ranking number ${\chi }_{\mathrm{r}}^{*}(G)$ of G is the corresponding on-line invariant; in that case vertices of G are coming one by one so that a partial ranking has to be chosen by considering only the structure of the subgraph of G induced by the present vertices. It is known that $\lfloor {\log }_{2}n\rfloor +1={\chi }_{\mathrm{r}}(P_n)⩽{\chi }_{\mathrm{r}}^{*}(P_n)⩽2\lfloor {\log }_{2}n\rfloor +1$. In this paper it is proved that ${\chi }_{\mathrm{r}}^{*}(P_n)>1.619{\log }_{2}n-1$.     

A sharp edge bound on the interval number of a graph

The interval number of a graph G, denoted by i(G), is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here we settle a conjecture of Griggs and West about bounding i(G) in terms of e, that is, the number of edges in G. Namely, it is shown that i(G) ≤ + 1. It is also observed that the edge bound induces i(G) ≤ , where γ(G) is the genus of G. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 153–159, 1999     

An upper bound for the harmonious chromatic number of a graph

An upper bound for the harmonious chromatic number of a graph G is given. Three corollaries of the theorem are theorems or improvements of the theorems of Miller and Pritikin.     

A lower bound for the equilateral number of normed spaces

We show that if the Banach-Mazur distance between an -dimensional normed space and is at most , then there exist equidistant points in . By a well-known result of Alon and Milman, this implies that an arbitrary -dimensional normed space admits at least equidistant points, where is an absolute constant. We also show that there exist equidistant points in spaces sufficiently close to , .

     

A lower bound for the convexity number of some graphs

Given a connected graphG, we say that a setC ?V(G) is convex inG if, for every pair of verticesx, y ∈ C, the vertex set of everyx-y geodesic inG is contained inC. The convexity number ofG is the cardinality of a maximal proper convex set inG. In this paper, we show that every pairk, n of integers with 2 ≤k ≤ n?1 is realizable as the convexity number and order, respectively, of some connected triangle-free graph, and give a lower bound for the convexity number ofk-regular graphs of ordern withn>k+1.     

A lower bound for the distance k-domination number of trees

A subset D of vertices of a graph G = (V, E) is a distance k-dominating set for G if the distance between every vertex of V ? D and D is at most k. The minimum size of a distance k-dominating set of G is called the distance k-domination number γ_k(G) of G. In this paper we prove that (2k + 1)γ_k(T) ≥ ¦V¦ + 2k ? kn₁ for each tree T = (V, E) with n₁ leafs, and we characterize the class of trees that satisfy the equality (2k + 1)γ_k(T) = ¦V¦ + 2k ? kn₁. Our results generalize those of Lemanska [4] for k = 1 and of Cyman, Lemanska and Raczek [1] for k = 2.