Let vt(d; 2) be the largest order of a vertex-transitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the Hoffman-Singleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The well-known degree/diameter problem asks for determining the largest possible number n(d; k) of vertic...

The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k . It is known that digraphs of order M d;k do not exist for d ? 1 and k ? 1 ([24] or [6]). In this paper we study digraphs of degree d, diameter k and order M d;k \Gamma 1, denoted by (d; k)-digraphs. Miller and Fris showed that (2; k)- digraphs do not exist for k 3 [22]. Subsequently, we gave a necessary condition of the existence of (3; k)-digraphs, namely, (3; k)-digraphs do not exist if k is odd or if k + 1 does not divide 9 2 (3 k \Gamma 1) [3]. The (d; 2)-digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of (d; k)-digraphs. In particular, for d; k 3, we show that a (d; k)-digraph contains either no cycle of length k or exactly one cycle of length k. 1 Introduction By a digraph we mean a structure G = (V; A) where V (G) is a nonempty set of elements called vertices; and A(G) is a set of ordered pairs (u; v) of disti...

It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [20] or [5]). Furthermore, for degree 2, it is shown that for k 3 there are no digraphs of order `close' to, i.e., one less than, Moore bound [18]. In this paper, we shall consider digraphs of diameter k, degree 3 and number of vertices one less than Moore bound. We give a necessary condition for the existence of such digraphs and, using this condition, we deduce that such digraphs do not exist for infinitely many values of the diameter. Keywords --- digraphs, Moore bound, diameter, degree. 1. Introduction By a digraph we mean a structure G = (V; A) where V (G) is a nonempty set of distinct elements called vertices; and A(G) is a set of ordered pairs (u; v) of distinct vertices u; v 2 V called arcs. The order of a digraph G is the number of vertices in G, i.e., jV (G)j. An inneighbour of a vertex v in a digraph G is a vertex u such that (u; v) 2 G. Similarly, an out-neighbour of a vertex v is a v...

It is known that Moore digraphs of degree d ? 1 and diameter k ? 1 do not exist (see [16] or [4]). For degree 2, it has been shown that for diameter k 3 there are no digraphs of order `close' to, i.e., one less than, the Moore bound [14]. For diameter 2, it is known that digraphs close to Moore bound exist for any degree because the line digraphs of complete digraphs are an example of such digraphs. However, it is not known whether these are the only digraphs close to Moore digraphs. In this paper, we shall consider the general case of directed graphs of diameter 2, degree d 3 and with the number of vertices n = d + d 2 , that is, one less than the Moore bound. Using the eigenvalues of the corresponding adjacency matrices we give a number of necessary conditions for the existence of such digraphs. Furthermore, for degree 3 we prove that there are no digraphs close to Moore bound other than the line digraph of K 4 . Keywords --- digraphs, Moore bound, diameter, degree. 1. Introduc...

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. The main objective of the paper is to revisit the classical degree/diameter problem for graphs from this new perspective. We derive a fairly general upper bound on the diameter of a lift in terms of the properties of the base voltage graph, and prove some results on vertex-transitive lifts. The potential of the new method is highlighted by showing that all currently known largest Cayley graphs (of given degree and diameter) for semidirect products of cyclic groups can be described by means of a voltage assignment construction, using simpler groups. This research started when J. Plesn'ik and J. Sir'an were visiting the Department of Computer Science of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. 1 Introduction In the past few decades there has been growing interest in the design of interconnection ...

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree # 15 and diameter # 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. # This research started when J. Plesnk and J. Siran were visiting the Department of Computer Science and Software Engineering of the University of Newcastle NSW Australia in 1995, supported by small ARC grant. the electronic journal of combinatorics 5 (1998), #R9 2 This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and diameter....

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...

The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k . It is known that digraphs of order M d;k do not exist for d ? 1 and k ? 1. In this paper we study digraphs of order M d;k \Gamma 1, that is, digraphs with defect 1, denoted by (d; k)-digraphs. If G is a (d; k)-digraph, then for each vertex v of G there exists a vertex w (called the repeat of v) such that there are two walks of lengths k from v to w. In the case of w = v we call v a selfrepeat. To study the existence of (d; k)-digraphs, we may divide the digraphs into two classes according to whether or not they contain a selfrepeat vertex. For d 3 and k 3 we prove that (d; k)-digraphs contain either no selfrepeats or exactly k selfrepeats. Furthermore, we show that every (d; k)-digraph with k selfrepeats must contain a cycle of length k as well as possibly another (d 1 ; k)-digraph as its subdigraph (where d 1 ! d). For diameter 2 we give further conditions for the existence of (d;...

The square G 2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree ∆(G) = 3 we have χ(G 2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of G 2 equals the chromatic number of G 2, that is χl(G 2) = χ(G 2) for all G. If true, this conjecture (together with Thomassen’s result) implies that every planar graph G with ∆(G) = 3 satisfies χl(G 2) ≤ 7. We prove that every graph (not necessarily planar) with ∆(G) = 3 other than the Petersen graph satisfies χl(G 2) ≤ 8 (and this is best possible). In addition, we show that if G is a planar graph with ∆(G) = 3 and girth g(G) ≥ 7, then χl(G 2) ≤ 7. Dvo˘rák, ˘ Skrekovski, and Tancer showed that if G is a planar graph with ∆(G) = 3 and girth g(G) ≥ 10, then χl(G 2) ≤ 6. We improve the girth bound to show that if G is a planar graph with ∆(G) = 3 and g(G) ≥ 9, then χl(G 2) ≤ 6. All of our proofs can be easily translated into linear-time coloring algorithms.