On traceable and upper traceable numbers of graphs

For a connected graph G of order n >= 2 and a linear ordering s: v(1), v(2), ..., v(n) of V(G), define d(s) = Sigma(n-1)(i=1) d(v(i), v(i+1)). The traceable number t(G) and upper traceable number t(+)(G) of G are defined by t(G) = min{d(s)} and t(+) (G) = max{d(s)}, respectively, where the minimum and maximum are taken over all linear orderings s of V(G). Consequently, t(G) <= t(+)(G). It is known that n - 1 <= t(G) <= 2n - 4' and n - 1 <= t(+)(G) <= left perpendicularn(2)/2left perpendicular - 1 for every connected graph G of order n >= 3 and, furthermore, for every pair n, A of integers with 2 <= n - 1 <= A <= 2n - 4 there exists a graph of order n whose traceable number equals A. In this work we determine all pairs A, B of positive integers with A <= B that are realizable as the traceable number and upper traceable number, respectively, of some graph. It is also determined for which pairs n,B of integers with n - 1 <= B <= left perpendicularn(2)/2left perpendicular - 1 there exists a graph whose order equals n and upper traceable number equals B.