# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2021): 1.028

5-year Journal Impact Factor (2021): 0.934

CiteScore (2021): 1.7

SNIP (2021): 1.157

# Discussiones Mathematicae Graph Theory

## VERTEX-DISTINGUISHING EDGE-COLORINGS OF LINEAR FORESTS

Sylwia Cichacz  and  Jakub Przybyło

AGH University of Science and Technology
Al. Mickiewicza 30, 30-059 Kraków, Poland
e-mail: przybylo@wms.mat.agh.edu.pl

## Abstract

In the PhD thesis by Burris (Memphis (1993)), a conjecture was made concerning the number of colors c(G) required to edge-color a simple graph G so that no two distinct vertices are incident to the same multiset of colors. We find the exact value of c(G) - the irregular coloring number, and hence verify the conjecture when G is a vertex-disjoint union of paths. We also investigate the point-distinguishing chromatic index, χ0(G), where sets, instead of multisets, are required to be distinct, and determine its value for the same family of graphs.

Keywords: irregular edge-coloring, vertex-distinguishing edge-coloring, point-distinguishing chromatic index.

2010 Mathematics Subject Classification: 05C15.

## References

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Received 12 May 2008
Revised 3 April 2009
Accepted 3 April 2009