Interval Total Colorings of Complete Multipartite Graphs and Hypercubes

TL;DR

This paper studies interval total colorings of complete multipartite graphs and hypercubes, establishing conditions for their colorability and bounds on coloring spans, advancing understanding of graph coloring properties.

Contribution

It proves that all complete multipartite graphs with equal parts are interval total colorable and characterizes interval total colorings of hypercubes, providing bounds on the span.

Findings

Complete multipartite graphs with equal parts are interval total colorable.

Hypercubes $Q_{n}$ have interval total $t$-colorings iff $n+1 \\leq t \\leq \\frac{(n+1)(n+2)}{2}$.

Bounds for the minimum and maximum span of these colorings are established.

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 interval total $t$-coloring of a graph $G$ is a total coloring of $G$ with colors $1,\ldots,t$ such that all colors are used, 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 a vertex $v$ in $G$. In this paper we prove that all complete multipartite graphs with the same number of vertices in each part are interval total colorable. Moreover, we also give some bounds for the minimum and the maximum span in interval total colorings of these graphs. Next, we investigate interval total colorings of hypercubes $Q_{n}$. In particular, we prove that $Q_{n}$ ($n\geq 3$) has an interval total $t$-coloring if and only if $n+1\leq t\leq \frac{(n+1)(n+2)}{2}$.