The 1, 2, 3-Conjecture states that the edges of a graph without isolated edges can be labeled from {1, 2, 3} so that the sums of labels at adjacent vertices are distinct. The 1, 2-Conjecture states that if vertices also receive labels and the vertex label is added to the sum of its incident edge labels, then adjacent vertices can be distinguished using only {1, 2}. We show that various configurations cannot occur in minimal counterexamples to these conjectures. Discharging then confirms the conjectures for graphs with maximum average degree less than 8/3. The conjectures are already confirmed for larger families, but the structure theorems and reducibility results are of independent interest.
@article{bwmeta1.element.doi-10_7151_dmgt_1768,
author = {Daniel W. Cranston and Sogol Jahanbekam and Douglas B. West},
title = {The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {769-799},
zbl = {1303.05172},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1768}
}
Daniel W. Cranston; Sogol Jahanbekam; Douglas B. West. The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 769-799. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1768/
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