Multigraphic Degree Sequences and Hamiltonian-connected Line Graphs

Let G be a multigraph. Suppose that e = u 1 v 1 and e ′ = u 2 v 2 are two edges of G. If e ≠ e ′, then G ( e, e ′) is the graph obtained from G by replacing e = u 1 v 1 with a path u 1 v e v 1 and by replacing e ′ = u 2 v 2 with a path u 2 v e′ v 2 , where v e , v e′ are two new vertices not in V ( G ). If e = e ′, then G ( e, e ′), also denoted by G( e ), is obtained from G by replacing e = u 1 v 1 with a path u 1 v e v 1 . A graph G is strongly spanning trailable if for any e, e ′ ∈ E ( G ), G ( e, e ′) has a spanning ( v e , v e ′ )-trail. The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties. A sequence d = ( d 1 , d 2 , ⋯, d n ) is multigraphic if there is a multigraph G with degree sequence d , and such a graph G is called a realization of d. A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph, and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected. In this paper, we prove that a nonincreasing multigraphic sequence d = ( d 1 , d 2 ) ⋯, d n ) is strongly spanning trailable if and only if either n = 1 and d 1 = 0 or n ≥ 2 and d n ≥ 3. Applying this result, we prove that for a nonincreasing multigraphic sequence d = ( d 1 , d 2 , ⋯, d n ), if n ≥ 2 and d n ≥ 3, then d is line-hamiltonian-connected.