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

Daniel W. Cranston; Sogol Jahanbekam; Douglas B. West

Discussiones Mathematicae Graph Theory (2014)

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

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topDaniel 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 -

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