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

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### 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.

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Combinatorics, Discrete mathematics & Graph theory