Even factors with a bounded number of components in iterated line graphs

We consider even factors with a bounded number of components in the n-times iterated line graphs Lⁿ(G). We present a characterization of a simple graph G such that Lⁿ(G) has an even factor with at most k components, based on the existence of a certain type of subgraphs in G. Moreover, we use this result to give some upper bounds for the minimum number of components of even factors in Lⁿ(G) and also show that the minimum number of components of even factors in Lⁿ(G) is stable under the closure operation on a claw-free graph G, which extends some known results. Our results show that it seems to be NP-hard to determine the minimum number of components of even factors of iterated line graphs. We also propose some problems for further research.