### Abstract

A (2,1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0,1,...,k} of nonnegative integers such that |f(x) - f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2,1)-total labeling number λ_{2}
^{T}(G) of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy λ_{2}
^{T}(G) < Δ(G) + 2, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ_{2}
^{T}(G) ≤ Δ(G) + 2 even in the case of Δ(G) ≤ 4.

Original language | English |
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Title of host publication | Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, Revised Selected Papers |

Pages | 103-106 |

Number of pages | 4 |

DOIs | |

Publication status | Published - Apr 4 2011 |

Event | 21st International Workshop on Combinatorial Algorithms, IWOCA 2010 - London, United Kingdom Duration: Jul 26 2010 → Jul 28 2010 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6460 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 21st International Workshop on Combinatorial Algorithms, IWOCA 2010 |
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Country | United Kingdom |

City | London |

Period | 7/26/10 → 7/28/10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, Revised Selected Papers*(pp. 103-106). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6460 LNCS). https://doi.org/10.1007/978-3-642-19222-7_11

**The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2.** / Hasunuma, Toru; Ishii, Toshimasa; Ono, Hirotaka; Uno, Yushi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, Revised Selected Papers.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6460 LNCS, pp. 103-106, 21st International Workshop on Combinatorial Algorithms, IWOCA 2010, London, United Kingdom, 7/26/10. https://doi.org/10.1007/978-3-642-19222-7_11

}

TY - GEN

T1 - The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2

AU - Hasunuma, Toru

AU - Ishii, Toshimasa

AU - Ono, Hirotaka

AU - Uno, Yushi

PY - 2011/4/4

Y1 - 2011/4/4

N2 - A (2,1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0,1,...,k} of nonnegative integers such that |f(x) - f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2,1)-total labeling number λ2 T(G) of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy λ2 T(G) < Δ(G) + 2, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ2 T(G) ≤ Δ(G) + 2 even in the case of Δ(G) ≤ 4.

AB - A (2,1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0,1,...,k} of nonnegative integers such that |f(x) - f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2,1)-total labeling number λ2 T(G) of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy λ2 T(G) < Δ(G) + 2, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ2 T(G) ≤ Δ(G) + 2 even in the case of Δ(G) ≤ 4.

UR - http://www.scopus.com/inward/record.url?scp=79953195503&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953195503&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-19222-7_11

DO - 10.1007/978-3-642-19222-7_11

M3 - Conference contribution

AN - SCOPUS:79953195503

SN - 9783642192210

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 103

EP - 106

BT - Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, Revised Selected Papers

ER -