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

Abstract

We consider even factors with a bounded number of components in the $n$-times iterated line graphs $L^{n}(G)$. We present a characterization of a simple graph $G$ such that $L^n(G)$ 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 $L^n(G)$ and also show that the minimum number of components of even factors in $L^n(G)$ 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.