A sharp upper bound for the harmonious total chromatic number of graphs and multigraphs

TL;DR

This paper establishes a tight upper bound for the harmonious total chromatic number of graphs and multigraphs, providing exact values for complete graphs and extending results to multigraphs with multiple edges.

Contribution

It proves the exact harmonious total chromatic number for complete graphs and derives a general upper bound for all graphs and multigraphs based on their order and maximum edge multiplicity.

Findings

Exact value of $h_t(K_n)$ for all $n$ except 1 and 4.

Upper bound $h_t(G) \\leq \\lceil 3/2 n ceil$ for graphs with more than 4 vertices.

Extension of bounds to complete multigraphs with multiple edges.

Abstract

A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest number of colours for it to exist is called the harmonious total chromatic number of $G$, denoted by $h_t(G)$. Here, we give a general upper bound for $h_t(G)$ in terms of the order $n$ of $G$. Our two main results are obvious consequences of the computation of the harmonious total chromatic number of the complete graph $K_n$ and of the complete multigraph $\lambda K_n$, where $\lambda$ is the number of edges joining each pair of vertices of $K_n$. In particular, Araujo-Pardo et al. have recently shown that $\frac{3}{2}n\leq h_t(K_n) \leq \frac{5}{3}n +\theta(1)$. In this paper, we prove that $h_t(K_{n})=\left\lceil \frac{3}{2}n \right\rceil$ except for $h_t(K_{1})=1$ and $h_t(K_{4})=7$; therefore, $h_t(G) \le \left\lceil \frac{3}{2}n \right\rceil$, for every graph $G$ on $n>4$ vertices. Finally, we extend such a result to the harmonious total chromatic number of the complete multigraph $\lambda K_n$ and as a consequence show that $h_t(\mathcal{G})\leq (\lambda-1)(2\left\lceil\frac{n}{2}\right\rceil-1)+\left\lceil\frac{3n}{2}\right\rceil$ for $n>4$, where $\mathcal{G}$ is a multigraph such that $\lambda$ is the maximum number of edges between any two vertices.