# Graphs with degree complete labeling

## TL;DR

This paper characterizes graphs that can be labeled to become degree complete, providing polynomial-time methods to recognize such graphs and find suitable labelings.

## Contribution

It offers three new characterizations of graphs with degree complete labelings and efficient algorithms for recognition and labeling.

## Key findings

- Three characterizations of graphs with degree complete labelings
- Polynomial-time algorithms for recognition and labeling
- Extension of Qian's characterization to unlabeled graphs

## Abstract

In 2006 Qian [J. Qian, Degree complete graphs; Discrete Mathematics 306 (2006), 533--537] introduced the concept of degree complete graphs for labeled graphs. He also gave a characterization of these graphs in terms of two forbidden subgraphs. Furthermore, he mentioned that the property of being degree complete depends on the labeling of the graph. Related to this he stated the problem to find a characterization of those (unlabeled) graphs for which every labeled version is not degree complete. We say that a (unlabeled) graph has a degree complete labeling, if there is a labeled version of the graph that is degree complete. In this paper we give three characterizations of graphs with degree complete labeling. These characterizations give us polynomial-time procedures to recognize these graphs and find a degree complete labeling, if it exists.