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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1479,
author = {Sylwia Cichacz and Jakub Przyby\l o},
title = {Vertex-distinguishing edge-colorings of linear forests},
journal = {Discussiones Mathematicae Graph Theory},
volume = {30},
year = {2010},
pages = {95-103},
zbl = {1215.05058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1479}
}
Sylwia Cichacz; Jakub Przybyło. Vertex-distinguishing edge-colorings of linear forests. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 95-103. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1479/
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