A note on the existence of factors in squares of graphs
Olga Fourtounelli, Panagiotis Katerinis · 2011
Let G be a simple connected graph such that δ(G) ≥ 3. For every function f: V (G) →{1, 2}, where ∑ x∈V (G) f(x) is even, the square graph G2 has an f-factor. All graphs considered are assumed to be simple and finite. We refer the reader to [2] for standard graph theoretic terms not defined in this paper. Let G be a graph. The degree dG(u) of a vertex u in G is the number of edges of G incident with u. The minimum degree of G is denoted by δ(G). If X and Y are subsets of V (G), we will write EG(X, Y)andeG(X, Y) for the set and the number, respectively, of the edges of G joining X to Y. For any set X of vertices in G, we define the neighbour set of X in G to be the set of all vertices adjacent to vertices in X; this set is denoted by NG(X). A set of vertices in G is said to be independent if no two of them are adjacent. The edge analogue of an independent set is a set of edges in G no two of which have a common