# Total proper connection and graph operations

## TL;DR

This paper investigates the total proper connection number in various graph operations, establishing that it is generally 3 for many combined graph types, including joins, products, and certain traceable graphs.

## Contribution

It determines the total proper connection number for several graph operations, showing it is typically 3 for joins, products, and specific traceable graphs, expanding understanding of graph coloring properties.

## Key findings

- Total proper connection number is 3 for joins, lexicographic, and strong products of nearly all graphs.
- The number is also 3 for Cartesian products with one traceable factor.
- Permutation graphs of stars and traceable graphs have a total proper connection number of 3.

## Abstract

A graph is said to be {\it total-colored} if all the edges and vertices of the graph are colored. A path in a total-colored graph is a {\it total proper path} if $(i)$ any two adjacent edges on the path differ in color, $(ii)$ any two internal adjacent vertices on the path differ in color, and $(iii)$ any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called {\it total-proper connected} if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph $G$, the {\it total proper connection number} of $G$, denoted by $tpc(G)$, is defined as the smallest number of colors required to make $G$ total-proper connected. In this paper, we study the total proper connection number for the graph operations. We find that $3$ is the total proper connection number for the join, the lexicographic product and the strong product of nearly all graphs. Besides, we study three kinds of graphs with one factor to be traceable for the Cartesian product as well as the permutation graphs of the star and traceable graphs. The values of the total proper connection number for these graphs are all $3$.