Abstract not found

We describe how standard backtracking rules of thumb were successfully applied to the problem of characterizing (3; g)- cages, the minimum order 3-regular graphs of girth g. It took just 5 days of cpu time (compared to 259 days for previous authors) to verify the (3; 9)-cages, and we were able to confirm that (3; 11)-cages have order 112 for the first time ever. The lower bound for a (3; 13)-cage is improved from 196 to 202 using the same approach. Also, we determined that a (3; 14)-cage has order at least 258. 1 Cages In this paper, we consider finite undirected graphs. Any undefined notation follows Bondy and Murty [7]. The girth of a graph is the size of a smallest cycle. A (r; g)- cage is an r-regular graph of minimum order with girth g. It is known that (r; g)-cages always exist [11]. Some nice pictures of small cages are given in [9, pp. 54-58]. The classification of the cages has attracted much interest amongst the graph theory community, and many of these have special nam...

. We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introduction Testing whether a graph has any axial (rotational, central, respectively) symmetry is a NP-complete problem [9]. Some restrictions (central symmetry with exactly one fixed vertex and no fixed edge) are polynomialy equivalent to the graph isomorphism test. Notice that this latter problem is not known to be either polynomial or NP-complete in general. But several heuristics are known (e.g. [3]) and several restrictions leads to efficient algorithms: linear time isomorphism test for planar graphs [6] and interval graphs [8], polynomial time isomorphism test for fixed genus [10, 5], k-contractible graphs [12] and pairwise k-separable graphs [11], linear axial symmetry detection for plana...

In the first part of this paper, we present a graphical construction that connects most of the known cubic cages. The construction suggests a technique for searching for new cages which we have used in seeking the 9-cage. In the second part, we describe the known cubic cages in another way, present a highly symmetric 58-vertex cubic graph with girth 9, and finally show how that graph fits into the partial order. 1 Definitions and Notation We will assume the usual machinery of graph theory. For a graph G; V (G) denotes the vertex set of G and E(G) denotes the edge set. The graphs we will consider will be simple, that is, they will not have loops (i.e., if e = fv 1 ; v 2 g 2 E(G) then v 1 6= v 2 ), nor will they have multiple edges. A cubic graph is simply a 3-regular graph: every vertex has degree three. We define a cubic tree to be a connected tree in which all vertices are either of degree three or of degree one. We will have particular use for a certain sequence of cubic trees t 0 ;...

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

Abstract not found