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

Abstract

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 $\delta(G)$ and $\eta(G)$ 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 $V_0\subseteq V(G)$ $G[V_0]$ denotes the subgraph of the graph $G$ induced by the subset $V_0$ of its vertices. An arbitrary nonempty finite subset of consecutive integers is called an interval. A function $\varphi:E(G)\rightarrow \{1,2,\dots,|E(G)|\}$ is called an edge labeling of the graph $G$, if for arbitrary different edges $e'\in E(G)$ and $e''\in E(G)$, the inequality $\varphi(e')\neq \varphi(e'')$ holds. If $G$ is a graph, $x$ is its arbitrary vertex, and $\varphi$ is its arbitrary edge labeling, then the set $S_G(x,\varphi)\equiv\{\varphi(e)/ e\in E(G), e \textrm{is incident with} x$\} is called a spectrum of the vertex $x$ of the graph $G$ at its edge labeling $\varphi$. If $G$ is a graph and $\varphi$ is its arbitrary edge labeling, then $V_{int}(G,\varphi)\equiv\{x\in V(G)/\;S_G(x,\varphi)\textrm{is an interval}\}$. For an arbitrary $r$-regular graph $G$ with $r\geq2$ and its arbitrary edge labeling $\varphi$, the inequality $$ |V_{int}(G,\varphi)|\leq\bigg\lfloor\frac{3\cdot|V(G)|-2\cdot\eta(G[V_{int}(G,\varphi)])}{4}\bigg\rfloor. $$ is proved.