A Caro-Wei bound for induced linear forests in graphs

A well-known result due to Caro (1979) and Wei (1981) states that every graph G has an independent set of size at least ∑v∈ V(G) 1/d(v) + 1, where $d(v)$ denotes the degree of vertex v. Alon, Kahn, and Seymour (1987) showed the following generalization: For every k≥ 0, every graph G has a k-degenerate induced subgraph with at least ∑v ∈ V(G)min\1, k+1/d(v)+1\ vertices. In particular, for $k=1$, every graph G with no isolated vertices has an induced forest with at least ∑v∈ V(G) 2/d(v) + 1 vertices. Akbari, Amanihamedani, Mousavi, Nikpey, and Sheybani (2019) conjectured that, if G has minimum degree at least $2$, then one can even find an induced linear forest of that order in G, that is, a forest where each component is a path. In this paper, we prove this conjecture and show a number of related results. In particular, if there is no restriction on the minimum degree of G, we show that there are infinitely many ``best possible'' functions f such that ∑v∈ V(G) f(d(v)) is a lower bound on the maximum order of a linear forest in G, and we give a full characterization of all such functions f.