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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

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