On odd primitive graphs

In this paper, we prove that every odd primitive graph must contain two vertex disjoint odd cycles. We also characterize a family of odd primitive graphs whose exponent achieves the upper bound. We follow the notation and terminology of Bondy and Murty [1], unless otherwise stated. A digraph D is said to be primitive if there exists a positive integer k such that for each ordered pair of vertices u, v there is a directed walk of length k from vertex u to vertex v in D. The smallest such integer k is called the exponent of D, denoted by 1(D). A primitive digraph is said to be odd primitive if its exponent is odd. It is well known that a digraph is primitive if and only if it is strongly connected and the greatest common divisor of the lengths of all its directed cycles is one. In this paper, we consider only symmetric digraphs without multiple edges, which we will call graphs. Let G be a graph. The odd girth of G is the length of a shortest odd cycle in G and is denoted by go(G). For two vertices u, v of G, we let 1CU, v) denote the smallest positive integer k such that there is a walk of length t from u to v in G for all t ;::: k. Obviously, if G is primitive, then 1CG) = maXu,vEV(G)'"Y(U, v). The basic properties of a primitive graph and its exponent given in the following propositions are well known. Proposition 1 [3]. A graph is primitive if and only if it is connected and contains an odd cycle. "'Project 1950115 supported by Natural Science Foundation of China and Shanxi Province. Australasian Journal of Combinatorics 19(1999). pp.11-15 Proposition 2 [2]. Let G be a primitive graph. If there are two walks from vertex u to vertex v with odd length kl and even length k2 respectively, then The following theorem on primitive graphs is due to J.Z. Wang and D.J. Wang [4]. Theorem 3 [4]. The set of exponents of all primitive graphs with order n and all odd girth go is {go 1, go, ... , 2n go 1} S where S is the set of zero and all odd integers s with n go + 1 ::; s ::; 2n go 1. Our objective in this paper is to study the structural properties of odd primitive graphs. For convenience, we give some further definitions and notation. The cartesian product X x Y of two graphs is the graph which has vertex-set V(X) x V(Y) and two vertices (Xl, Yl), (X2' Y2) of X x Y adjacent whenever either Xl = X2 and YIY2 E E(Y) or Yl = Y2 and XIX2 E E(X). A v v'-lollipop is the graph obtained by taking the union of a cycle C and a v v' path P such that C n P = {v'}. Clearly, PuC uP is a walk from v to v. For a walk W, we will denote its length by l(W). Our main results are the following theorems. Theorem 4. Let G be an odd primitive graph. Then G contains two vertex disj.oint odd cycles. Proof. Assume that G is an odd primitive graph. Then there exists a pair of vertices u, v such that ')'( G) = ')'( u, v) and thus there are no walks of length ')'( G) -1 from u to v in G. Let Wu and Wv be walks of the shortest odd lengths from vertex u to vertex u and from vertex v to vertex v, respectively. Then the lengths of Wu and Wv are both not greater that ')'( G). Since any walk of odd length from a vertex of G to itself must contain an odd cycle, we can assume that Wu = Pu U Cu U Pu is an u u' -lollipop and Wv Pv U Cv U Pv is a v v'-lollipop. Since both l(Wu) and l(Wv) are odd, Cu and Cv are two odd cycles in G. Next, we show that Cu n Cv = 0. Otherwise, let x E Cu n Cv. If Cu and Cv are the same cycle, then u' and v' divide Cu into two parts, C~ and C~ say (possibly, u' = v') and so Pu U C~ U Pv and Pu U C~ U Pv are two walks from vertex u to v whose lengths have different parity. We apply Proposition 2 to obtain that ')'( u, v) < max{l(Pu U C~ U Pv), l(Pu U C; U Pv)} 1 < l(Pu U Cu U Pv) 1 < I(Wu ) + l(Wv) _ 1 2 < ')'(G) 1, contradicting the assumption. Hence Cu =f. Cv' Then x and u' divide Cu into two parts C~ and C~, x and v' divide Cv into two parts C~ and C~. It is seen easily that