Long heterochromatic paths in heterochromatic triangle free graphs

TL;DR

This paper investigates the existence of long heterochromatic paths in heterochromatic triangle free graphs, providing bounds based on vertex color degrees for complete and general graphs.

Contribution

It establishes new lower bounds on heterochromatic path lengths in heterochromatic triangle free graphs with specific coloring conditions.

Findings

In complete heterochromatic triangle free graphs, each vertex has a heterochromatic path of length at least its color degree.

In general heterochromatic triangle free graphs with minimum color degree at least 6, there exists a heterochromatic path of length at least 75% of that degree.

The results extend understanding of heterochromatic paths in edge-colored graphs with forbidden triangles.

Abstract

In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic triangle free complete graphs; the other is heterochromatic triangle free graphs with $k$-good colorings, i.e., minimum color degree at least $k$. For the heterochromatic triangle free graphs $K_n$, we obtain that for every vertex $v\in V(K_n)$, $K_n$ has a heterochromatic $v$-path of length at least $d^c(v)$; whereas for the heterochromatic triangle free graphs $G$ we show that if, for any vertex $v\in V(G)$, $d^c(v)\geq k\geq 6$, then $G$ a heterochromatic path of length at least $\frac{3k}{4}$.