Iterated line graphs are maximally ordered

Abstract

Abstract:A graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle in G that encounters the vertices of the sequence in the given order. We prove that if G is a connected graph distinct from a path, then there is a number t G such that for everySince there is no graph H which is (δ(H)+2)-ordered, the result is best possible.

The existence of even factors in iterated line graphs

“…Xiong and Liu [11] characterized the graphs with a hamiltonian n-iterated line graphs, while Ferrara, Gould and Hartke [2] gave a characterization of the graphs whose n-iterated line graphs have a 2-factor. Some other properties on iterated line graphs have appeared, such as k-orderability [5] and linkability [1]. In this paper, we will give a characterization of the graphs G such that L n (G) has an even factor, which is different from that above as we do not consider the existence of a certain type of subgraph in G. A branch is a nontrivial path whose internal vertices have degree 2 and end vertices have degree other than 2.…”

The structure and existence of 2-factors in iterated line graphs

“…Chartrand [2] was one of the first to study properties of iterated line graphs, proving that for every graph G (with a few trivial exceptions), L k (G) is hamiltonian for k sufficiently large. Since this first paper, many cyclestructural properties of iterated line graphs have been studied, including when L k (G) is k-ordered ( [10]), pancyclic ( [12]), k-ordered hamiltonian ( [8]), and characterizations of G when L k (G) is hamiltonian ( [11]).…”