2024/03/26 by Gwenaël Joret, Joret, Gwenaël, Robin Petit +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2403.17568
openalex publication_date 2024/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, \frac k+1d(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.