VERTEX EXPONENTS OF A CLASS OF TWO-COLORED HAMILTONIAN DIGRAPHS
Abstract: A two-colored
digraph D(2) is primitive provided there are nonnegative integers h and k such
that for each pair of not necessarily distinct vertices u and v inD(2) there
exists a (h; k)-walk in D(2) from u to v. The exponent of a primitive
twocolored digraph D(2), exp(D(2)), is the smallest positive integer h + k
taken over all such nonnegative integers h and k. The exponent of a vertex v in
D(2) is the smallest positive integer s + t such that for each vertex u in D(2)
there is an (s; t)-walk from v to u. We study the vertex exponents of primitive
two-colored digraphs L(2) n on n ≥ 5 vertices whose underlying digraph is the
Hamiltonian digraph consisting ofthe cycle v1 ! vn ! vn−1 ! · · · ! v2 ! v1 and
the arc v1 ! vn−2. For such two-colored digraph it is known that 2n2 − 6n + 2 ≤
exp(L(2) n ) ≤ (n3 − 2n2 + 1)=2. We show that if exp(L(2) n ) = (n3 − 2n2 +
1)=2, then its vertex exponents lie on[(n3 − 2n2 − 3n + 4)=4; (n3 − 2n2 + 3n +
6)=4] and if exp(L(2) n ) = 2n2 − 6n + 2, then its vertex exponents lie on [n2
− 4n + 5; n2 − 2n − 1].
Author: Aghni Syahmarani and
Saib Suwilo
Journal Code: jpmatematikagg120008
