Spanning connectedness and Hamiltonian thickness of graphs and interval graphs - Peng Li; Yaokun Wu (Discrete Mathematics & Theoretical Computer Science, 2015)
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 = π1 · · · πn 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 πiπi+1 ∈ E(G) and the number of neighbors of πi among {πi+1,. .. , πn} 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 : πiπi+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.