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

TL;DR

The paper determines the exact harmonious total chromatic numbers for complete graphs, showing $h_t(K_n)=\left\lceil\frac{3n}{2}\right\rceil$ for all $n$ except $n=1$ and $n=4$ (with $h_t(K_1)=1$ and $h_t(K_4)=7$). It then provides a constructive, case-split proof that achieves this bound for all $n>4$ and extends the results to complete multigraphs, establishing $h_t(\lambda K_n)=(\lambda-1)(2\left\lceil n/2\right\rceil-1)+\left\lceil\frac{3n}{2}\right\rceil$ (with $h_t(\lambda K_4)=3\lambda+4$). The approach relies on explicit vertex-labelings, 1-factorizations, and careful edge-colouring arguments, including Cayley-graph and Vizing-type techniques, to produce tight, constructive colourings. Consequently, the paper provides a tight general upper bound $h_t(G)\le\left\lceil\frac{3n}{2}\right\rceil$ for graphs on $n>4$ vertices and a precise framework for harmonious total colourings of dense graphs and multigraphs.

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 $λK_n$, where $λ$ is the number of edges joining each pair of vertices of $K_n$. In particular, Araujo-Pardo et al. have recently shown that $\frac{3}{2}n\leq h_t(K_n) \leq \frac{5}{3}n +θ(1)$. 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 $λK_n$ and as a consequence show that $h_t(\mathcal{G})\leq (λ-1)(2\left\lceil\frac{n}{2}\right\rceil-1)+\left\lceil\frac{3n}{2}\right\rceil$ for $n>4$, where $\mathcal{G}$ is a multigraph such that $λ$ is the maximum number of edges between any two vertices.

Figures (2)

• Figure 1: Harmonious total colourings of $K_8$ with 12 colours.
• Figure 2: Harmonious total colourings of some small complete graphs.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Corollary 5
• proof