# The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs

• Volume: 34, Issue: 4, page 769-799
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Daniel W. Cranston, Sogol Jahanbekam, and Douglas B. West. "The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs." Discussiones Mathematicae Graph Theory 34.4 (2014): 769-799. <http://eudml.org/doc/269827>.

@article{DanielW2014,
abstract = {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.},
author = {Daniel W. Cranston, Sogol Jahanbekam, Douglas B. West},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {1; 2; 3-Conjecture; 1; 2-Conjecture; reducible configuration; discharging method.; -conjecture; -conjecture; discharging method},
language = {eng},
number = {4},
pages = {769-799},
title = {The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs},
url = {http://eudml.org/doc/269827},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Daniel W. Cranston
AU - Sogol Jahanbekam
AU - Douglas B. West
TI - The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2014
VL - 34
IS - 4
SP - 769
EP - 799
AB - 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.
LA - eng
KW - 1; 2; 3-Conjecture; 1; 2-Conjecture; reducible configuration; discharging method.; -conjecture; -conjecture; discharging method
UR - http://eudml.org/doc/269827
ER -

## References

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