Digraph covering and its application to two optimization problems

as In an 2k+l values The paper concludes with the N(d, k) an;-alSJI[)HJ~i and circuit cove1'ings then show that certain fi1.1'(Yl1lll1'.r We introduce shows This new result also our present the values d and n 5.100. INTRODUCTION A directed or 'Unl"llllll. G where (i) II is nonempty set 11 , •• 1 v n1 , •• } of distinct elements called vertices; (ii) E is called arcs. E = {el' e2, ... 1 em,"'} of ordered In this paper we shall consider 'U.A/"hU,t-' ... "" in which both 11 and E are finite. The number of vprt.l,(,(";: in the digraph. The indegree of a vertex Similarly, the outdegree of The degree of a vertex v is every vertex has the same regular digraph of degree d. number of the form (u,v) in G. u EGis the number of arcs of the form (u, v) in ,..,,.'po-.,.,,,,p and If in a digraph d then G dig1'aph, or a Australasian Journal of Combinatorics 3(1991) pp 151-164 However, if G in G then G a digraph in which 1n{iPC'rp,P=r.lltCl.(~gr'ee:=a for every vertex called a diregular digraph of degree d. The diameter k of vertices of G. C1.H~ra·Ph G the maximum shortest distance between any two A diregular digraph G of order n, d and diameter k, G E 9(n, d, k) consists of n vertices and nd arcs. For each vert.ex v G there are d arcs whose initial vertex is v, and d arcs whose terminal vertex is v. Given a digraph G, set II 1 12)"" lm of subdigraphs of G covers G if every arc and every vertex of G OCCurs in some li. and a collection lil 1 i m of subdigraphs of G such that. U~lli G is a covering of G. We introduce the following definitions. Definition 1. A covering L of G is arc-disjoint if for all Ii and lj in L, li and lj have no arcs in common whenever i ;. j. Similarly, we can define the concept of a vertex-disjoint covering. Definition A L,UVCI"H1t'L of a digraph G is a circuit covering if for all Ii E L, the subdigraph It is a circuit. Furthermore, we call a circuit covering consisting of circuits of lengths t 1 , t 2 , ••• ,ts a t 1 ,. ., is-circuit covering. In particular, a icircuit covering consists only of circuits of length t. Lastly, a ::; t-circuit covering consists of circuits of length t or less, and a i-circuit covering consists of circuits of lengths less than t. The three optimization problems N(d,k), K(n,d) and D(n,k) for diregular digraphs 9(n, d, k) are defined as follows. 1. The N(d, k) problem: given d and k, find the maximum possible order N(d, k). This problem has been also called the (d, k ) problem. 2. The K( n, d) problem: given nand d, find the minimum possible diameter K(n, 3. The D(n,k) problem given nand k, find the minimum possible degree D(n,k). These three problems are related, but as far as we know, not equivalent. For further discussion on the relationships between these problems see [5]. The main result of this paper, Theorem 3, namely that for many values of k, 9(n, 2, k) empty, gives new results for the first two problems for d = 2. The proofs of all the theorems in this paper make use of circuit coverings. Every diregular digraph G E 9( n, d, k) has a ~ k + l-circuit covering since every vertex must reach itself in at most k + 1 steps (in at least d ways).