Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using any local search algorithm as a subroutine. Its effectiveness is illustrated by solving several classical combinatorial or global optimization problems. Moreover, several extensions are proposed for solving large problem instances: using VNS within the successive approximation method yields a two-level VNS, called variable neighborhood decomposition search (VNDS); modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed VNS (SVNS) heuristic. Finally, we show how to stabilize column generation algorithms with help of VNS and discuss various ways to use VNS in graph theory, i.e., to suggest, disprove or give hints on how to prove conjectures, an area where metaheuristics do not appear

Graph theory owes many powerful ideas and constructions to geometry. Several well-known families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.

We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

The largest known 3-regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3-regular, diameter-3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)-graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)-graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)-graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...

Let G(v, δ) be the set of all δ-regular graphs on v vertices. Certain graphs from among those in G(v, δ) with maximum girth have a special property called trace-minimality. In particular, all strongly regular graphs with no triangles and some cages are trace-minimal. These graphs play an important role in the statistical theory of D-optimal weighing designs. Each weighing design can be associated with a (0, 1)-matrix. Let Mm,n(0, 1) denote the set of all m × n (0,1)-matrices and let G(m, n) = max{det X T X: X ∈ Mm,n(0, 1)}. A matrix X ∈ Mm,n(0, 1) is a D-optimal design matrix if det X T X = G(m, n). In this paper we exhibit some new formulas for G(m, n) where n ≡ −1 (mod 4) and m is sufficiently large. These formulas depend on the congruence class of m (mod n). More precisely, let m = nt + r where 0 ≤ r < n. For each pair n, r, there is a polynomial P (n, r, t) of degree n in t, which depends only on n, r, such that G(nt + r, n) = P (n, r, t) for all sufficiently large t. The polynomial P (n, r, t) is computed from the characteristic polynomial of the adjacency matrix of a trace-regular graph whose degree of regularity and number of vertices depend only on n and r. We obtain explicit expressions for the polynomial P (n, r, t) for many pairs n, r. In particular we obtain formulas for G(nt + r, n) for n = 19, 23, and 27, all 0 ≤ r < n, and all sufficiently large t. And we obtain families of formulas for P (n, r, t) from families of trace-minimal graphs including bipartite graphs obtained from finite projective planes, generalized quadrilaterals, and generalized hexagons. Keywords: D-optimal weighing design, trace-minimal graph, regular graph, strongly regular graph, girth, cages, generalized polygons AMS Subject Classification:

Some properties of G(8, 3) are presented showing its uniqueness among generalized Petersen graphs.