### Spanning connectedness and Hamiltonian thickness of graphs and interval graphs

*Peng Li, Yaokun Wu*

#### Abstract

A spanning connectedness property is one which involves the robust
existence of a spanning subgraph which is of some special form, say a
Hamiltonian cycle in which a sequence of vertices appear in an
arbitrarily given ordering, or a Hamiltonian path in the subgraph
obtained by deleting any three vertices, or three
internally-vertex-disjoint paths with any given endpoints such that
the three paths meet every vertex of the graph and cover the edges of
an almost arbitrarily given linear forest of a certain fixed size.
Let π=π

_{1}⋯π_{n}be an ordering of the vertices of an n-vertex graph G. For any positive integer k≤n-1, we call π a k-thick Hamiltonian vertex ordering of G provided it holds for all i∈{1, …, n-1} that π_{i}π_{i+1}∈E(G) and the number of neighbors of π_{i}among {π_{i+1}, …, π_{n}} is at least min(n-i, k); For any nonnegative integer k, we say that π is a -k-thick Hamiltonian vertex ordering of G provided |{i: π_{i}π_{i+1}∉E(G), 1≤i≤n-1}|≤k+1. Our main discovery is that the existence of a thick Hamiltonian vertex ordering will guarantee that the graph has various kinds of spanning connectedness properties and that for interval graphs, quite a few seemingly unrelated spanning connectedness properties are equivalent to the existence of a thick Hamiltonian vertex ordering. Due to the connection between Hamiltonian thickness and spanning connectedness properties, we can present several linear time algorithms for associated problems. This paper suggests that much work in graph theory may have a spanning version which deserves further study, and that the Hamiltonian thickness may be a useful concept in understanding many spanning connectedness properties.Full Text: PDF