The upper traceable number of a graph

On upper traceable numbers of graphs

“…By Theorem 1.2, the upper traceable number of a nontrivial tree is always odd. With the aid of Theorem 1.2, sharp upper and lower bounds for the upper traceable number of a tree were established in [9] in terms of its order, as we state now. Theorem 1.3 [9].…”

“…Furthermore, t(G) = n − 1 if and only if G is traceable. In fact, the traceable number of a connected graph G is the minimum length of a spanning walk in G. The Hamiltonian number h(G) and traceable number t(G) of a graph G provide measures of traversability for G. For a connected graph G, the upper Hamiltonian number h + (G) is defined in [5] as h + (G) = max {d(s)}, where the maximum is taken over all cyclic orderings s of vertices of G. As expected, for a connected graph G, the upper traceable number t + (G) is defined in [9] as…”

“…For each edge e of a tree T , the component number cn(e) of e is defined in [5] as the minimum order of a component of T − e. For example, the edge e 8 of the tree T of Figure 2 The following result provides a formula for the upper traceable number of a tree in terms of the component numbers of its edges. Theorem 1.2 [9]. If T is a nontrivial tree, then…”

“…Both upper and lower bounds in (2) are sharp. Characterizations of all graphs whose upper traceable number and traceable number differ by at most 1 have been established in [9]. Theorem 1.1 [9].…”

“…Let G be a connected graph of order n 3. Then The upper traceable numbers of some well-known classes of graphs (namely, complete multipartite graphs, cycles, hypercubes) have been determined (see [9]). In particular, a formula for the upper traceable number of a tree was established.…”