Estimates for the number of vertices with an interval spectrum in proper edge colorings of some graphs

TL;DR

This paper provides estimates on the number of vertices with interval spectra in proper edge colorings of certain graphs, advancing understanding of coloring properties relevant to graph theory.

Contribution

It introduces bounds on the count of vertices with interval or persistent-interval spectra in proper edge colorings for specific graph classes.

Findings

Estimates for vertices with interval spectra in certain graphs.

Bounds on vertices with persistent-interval spectra.

Application to classes of graphs with specific coloring properties.

Abstract

A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that each of $t$ colors is used, and adjacent edges are colored differently. The set of colors of edges incident with a vertex $x$ of $G$ is called a spectrum of $x$. A proper edge $t$-coloring of a graph $G$ is interval for its vertex $x$ if the spectrum of $x$ is an interval of integers. A proper edge $t$-coloring of a graph $G$ is persistent-interval for its vertex $x$ if the spectrum of $x$ is an interval of integers beginning from the color 1. For graphs $G$ from some classes of graphs, we obtain estimates for the possible number of vertices for which a proper edge $t$-coloring of $G$ can be interval or persistent-interval.