A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

TL;DR

This paper provides a shorter proof of the Bang-Jensen-Gutin conjecture on Hamilton paths in edge-colored complete graphs, introduces a new condition for properly-colored cycles, and derives Yeo's theorem using a closure concept.

Contribution

It offers a concise proof of the conjecture, a novel sufficient condition for properly-colored cycles, and a new derivation of Yeo's theorem.

Findings

Shorter proof of Bang-Jensen-Gutin conjecture

New sufficient condition for properly-colored cycles

Derivation of Yeo's theorem from the new condition

Abstract

A path (cycle) is properly-colored if consecutive edges are of distinct colors. In 1997, Bang-Jensen and Gutin conjectured a necessary and sufficient condition for the existence of a Hamilton path in an edge-colored complete graph. This conjecture, confirmed by Feng, Giesen, Guo, Gutin, Jensen and Rafley in 2006, was laterly playing an important role in Lo's asymptotical proof of Bollob\'as-Erd\H{o}s' conjecture on properly-colored Hamilton cycles. In 1997, Yeo obtained a structural characterization of edge-colored graphs that containing no properly colored cycles. This result is a fundamental tool in the study of edge-colored graphs. In this paper, we first give a much shorter proof of the Bang-Jensen-Gutin Conjecture by two novel absorbing lemmas. We also prove a new sufficient condition for the existence of a properly-colored cycle and then deduce Yeo's theorem from this result and a closure concept in edge-colored graphs.