An inequality for the number of vertices with an interval spectrum in edge labelings of regular graphs

We consider undirected simple finite graphs. The sets of vertices and edges of a graph G are denoted by V (G) and E(G), respectively. For a graph G, we denote by δ(G) and η(G) the least degree of a vertex of G and the number of connected components of G, respectively. For a graph G and an arbitrary subset V0 ⊆ V (G) G[V0] denotes the subgraph of the graph G induced by the subset V0 of its vertices. An arbitrary nonempty finite subset of consecutive integers is called an interval. A function φ : E(G) → {1, 2, . . . , |E(G)|} is called an edge labeling of the graph G, if for arbitrary different edges e ∈ E(G) and e ∈ E(G), the inequality φ(e) 6= φ(e) holds. If G is a graph, x is its arbitrary vertex, and φ is its arbitrary edge labeling, then the set SG(x, φ) ≡ {φ(e)/e ∈ E(G), e is incident with x} is called a spectrum of the vertex x of the graph G at its edge labeling φ. If G is a graph and φ is its arbitrary edge labeling, then Vint(G,φ) ≡ {x ∈ V (G)/ SG(x, φ) is an interval}. For an arbitrary r-regular graph G with r ≥ 2 and its arbitrary edge labeling φ, the inequality |Vint(G,φ)| ≤ ⌊ 3 · |V (G)| − 2 · η(G[Vint(G,φ)]) 4 ⌋ . is proved.