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

Abstract

We prove several results about the structure of 2-factors in it- erated line graphs. Specifically, we give degree conditions on G that ensure L 2 (G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor inL 2 (G) with all cycle lengths specified. We also give a characterization of the graphs G where L k (G) A spanning cycle in a graph G is called a hamiltonian cycle and if such a cycle exists, we say that G is hamiltonian. Similarly, a 2-factor in a graph G is a 2-regular spanning subgraph, or equivalently a partition of V (G) into cycles. Hamiltonicity and the existence of 2-factors in graphs have been widely studied. A good reference for the current state of such problems is (4). We say that two edges in a graph G are adjacent if they share an end-vertex. The line graph of G, denoted L(G) is the graph with V (L(G)) = E(G) and E(L(G)) = {eiej | ei and ej are adjacent in G}. We define the i th iterated line graph of G recursively with L 1 (G) = L(G) and L i+1 (G) = L(L i (G)). 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 cycle-structural 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)). In this paper, we extend Chartrand's result by giving degree conditions on G to ensure that L 2 (G) contains a 2-factor with every possible number of cycles. We also give a sufficient condition for the existence of a 2-factor in L 2 (G) with all cycle lengths specified. Finally in section 5, we give a characterization of the graphs G where L k (G) contains a 2-factor. 2. Preliminaries In the majority of this paper, we consider only undirected, loopless graphs with- out multiple edges. The main results in the last section will consider multigraphs as well. We will use nk to denote the number of vertices in the k th iteration of the line graph, andk to denote �(L k (G)).