Interval total colorings of graphs

TL;DR

This paper studies interval total colorings of graphs, exploring their properties and determining the minimum and maximum number of colors needed for specific graph classes.

Contribution

It introduces the concept of interval total colorings and provides exact values for the range of colors for certain graph classes.

Findings

Characterization of interval total colorings

Exact values for minimum and maximum colors in specific graphs

Properties and bounds of interval total colorings

Abstract

A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An \emph{interval total $t$-coloring} of a graph $G$ is a total coloring of $G$ with colors $1,2,\...,t$ such that at least one vertex or edge of $G$ is colored by $i$, $i=1,2,\...,t$, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this paper we investigate some properties of interval total colorings. We also determine exact values of the least and the greatest possible number of colors in such colorings for some classes of graphs.