A note on degree-constrained subgraphs

Elementary proofs are presented for two graph theoretic results, originally proved by H. Shirazi and J. Verstraëte using the combinatorial Nullstellensatz. In an undirected graph G = (V, E) we denote by dG(v) the degree of v ∈ V . If F (v) ⊆ N is a set of forbidden degrees for every v ∈ V , then a subgraph G = (V, E) of G is called F -avoiding if dG′(v) / ∈ F (v) for all v ∈ V . Theorem 1 (Shirazi, Verstraëte [5]). If G = (V, E) is an undirected graph and |F (v)| ≤ dG(v)/2 for every node v, (1) then G has an F -avoiding subgraph. Theorem 1 appeared first under the name Louigi’s conjecture in [1]. A version with dG(v)/2 replaced by dG(v)/12 was given in [1], while dG(v)/8 was proved in [2]. Louigi’s conjecture was first settled in the affirmative by H. Shirazi and J. Verstraëte [5]. Their proof is based on the combinatorial Nullstellensatz of N. Alon [3]. We give an elementary proof, which uses Theorem 2 below. In a directed graph D = (V, − → E ) we denote by %D(v) the in-degree of v ∈ V . Theorem 2. If G = (V, E) is an undirected graph and it has an orientation D for which %D(v) ≥ |F (v)| for every node v, then G has an F -avoiding subgraph. Proof. For an undirected edge e, let − →e denote the corresponding directed edge of D. We use induction on the number of edges. If 0 is not a forbidden degree at any node, then the empty subgraph (V, ∅) is F -avoiding. Suppose that 0 ∈ F (t) for a node t. Then %D(t) ≥ |F (t)| ≥ 1 and hence there is an edge e = st of G for which − →e is directed toward t. Let G = G − e and D = D − − →e . Define MTA-ELTE Egerváry Research Group, Department of Operations Research, Eötvös University, Pázmány P. s. 1/C, Budapest, Hungary, H-1117. e-mail: {frank, jacint} @cs.elte.hu. Supported by the Hungarian National Foundation for Scientific Research, OTKA K60802, TS 049788, and by European MCRTN Adonet, Contract Grant No. 504438. Department of Computer Science and Engineering, The Chinese University of Hong Kong. e-mail: [email protected]. Research was done while the author visited the EGRES Group. Supported by European MCRTN Adonet, Contract Grant No. 504438.