The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs
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 lab...
| Published in: | Discussiones Mathematicae Graph Theory |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
University of Zielona Góra
2014-11-01
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| Subjects: | |
| Online Access: | https://doi.org/10.7151/dmgt.1768 |
| _version_ | 1852790512281452544 |
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| author | Cranston Daniel W. Jahanbekam Sogol West Douglas B. |
| author_facet | Cranston Daniel W. Jahanbekam Sogol West Douglas B. |
| author_sort | Cranston Daniel W. |
| collection | DOAJ |
| container_title | Discussiones Mathematicae Graph Theory |
| description | 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. |
| format | Article |
| id | doaj-art-2bbdbbca8cee4ef2b4269b0a402133ed |
| institution | Directory of Open Access Journals |
| issn | 2083-5892 |
| language | English |
| publishDate | 2014-11-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| spelling | doaj-art-2bbdbbca8cee4ef2b4269b0a402133ed2025-08-19T20:44:01ZengUniversity of Zielona GóraDiscussiones Mathematicae Graph Theory2083-58922014-11-0134476979910.7151/dmgt.1768dmgt.1768The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse GraphsCranston Daniel W.0Jahanbekam Sogol1West Douglas B.2Virginia Commonwealth University Richmond, VA, USAUniversity of Colorado Denver Denver, CO, USAZhejiang Normal University, Jinhua, China and University of Illinois, Urbana, IL, USAThe 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.https://doi.org/10.7151/dmgt.1768123-conjecture12-conjecturereducible configurationdischarging method. |
| spellingShingle | Cranston Daniel W. Jahanbekam Sogol West Douglas B. The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs 1 2 3-conjecture 1 2-conjecture reducible configuration discharging method. |
| title | The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs |
| title_full | The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs |
| title_fullStr | The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs |
| title_full_unstemmed | The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs |
| title_short | The 1 , 2 , 3-Conjecture And 1 , 2-Conjecture For Sparse Graphs |
| title_sort | 1 2 3 conjecture and 1 2 conjecture for sparse graphs |
| topic | 1 2 3-conjecture 1 2-conjecture reducible configuration discharging method. |
| url | https://doi.org/10.7151/dmgt.1768 |
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