The maximum degree of a critical graph of diameter two

A graph of diameter k is critical if the deletion of any vertex results in a of diameter at least k + 1. It was conjectured by Boals et. al. that the maximum of a critical graph of diameter 2 on n vertices is at most the that for n 2 7 the maximum in such a graph is n 4 when n is odd and n 5 when n is even. We construct for each n 2: 7 a critical graph of maximum degree Our graphs provide a good lower bound for the maximum number of possible in such a graph. We that for any integers nand p with n 2 7 and + 3)} ~ p ::; there exists a critical graph on n vertices of diameter 2 and maximum p. This that for any p 2:: 5 there exists a critical graph of diameter 2 and maximum degree p on lHp2 + 4p l)J vertices. Modern communication networks often face message delay problems caused by the unavailability (through failure or occupancy) of its components or junctions. Graph theory concepts are useful in analyzing the efficiency and reliability of such networks. In many network applications, the problem that arises is to construct a network which satisfies certain and which is optimal according to some criterion such as output, or theory is particularly useful when the requirements of the network can be in terms of graph parameters. For example, the diameter d( G) of a graph which is defined as the maximum distance in G, provides a measure of the efficiency of the underlying network; d( G) is just the maximum number of links needed to connect any two points (components) in the network. Thus studying graph parameters can provide useful information. In this paper, we focus on the parameter d(G). In characterizing graphs with prescribed diameter, it is fruitful to consider a subclass of graphs, the so called critical graphs. Australasian Journal of Combinatorics 16(1997), pp.245-258 Let G be a connected graph. The distance, de (u, v), between the vertices u and v in G is the length of a shortest (u, v)-path in G. The diameter d(G) is thus d(G) = max{de(u, v): u, v E V(G)} A vertex v of G is critical if d(G v) > d(G). If every vertex of G is critical, then G is called critical. Critical graphs have been studied in [1] [6]. Gliviak [4] observed that a graph G with minimum degree at least two and (the length of a shortest cycle) at least d( G) + 3 is a critical graph. A well-known result is that every critical graph is a block [1, 4, 6]. Critical graphs of small diameters have received particular attention. In this paper, we mainly consider those of diameter 2. Denote by 9(n,p) the class of all critical graphs on n vertices of diameter 2 and maximum degree p. For simplicity, we shall refer to each graph in 9 (n, p) as an (n, p) -critical graph. One natural question to ask is: For which values of nand p, is 9 (n, p) f. 0? Boals et. al. [1] conjectured that 9 (n, p) = 0 for all p > An affirmative answer to the conjecture would imply that every critical graph contains at most ~n2 edges. Unfortunately the conjecture is false. For n :::::: 7, we show that the maximum number for which 9 (n, .6.n ) f. 0 is n 4 when n is odd and n 5 when n is even. In general, we show, by construction, that 9(n,p) f. 0 for all nand p with n :::::: 7 and min{ J4n + 5 2, i(n + 3)} :::; p:::; .6.n . Let G be a graph. Two adjacent vertices x and y will be denoted by x rv y. For each vertex v, we shall use Li ( v) (i :::::: 1) to denote the set of all vertices which have distance i from v and shall call each L i ( v) a distance level of v. Suppose that G is a graph of 9 (n, p). Let L1 (v), L2 (v), ... , Lk (v) be the non-empty distance levels of a vertex v. Then we must have k = 2. Thus for each vertex u f. v, u E L1(V) U L2(V). Since u is critical, there are two vertices x and y such that x rv U rv y is the only path of length ~ 2 joining x and y. This implies that x and yare not adjacent and at least one of them must be in L2 ( v ). This fact will be frequently used throughout our proofs.