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 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.
| Original language | English |
|---|---|
| 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) |
|---|---|
| Volume | 6460 LNCS |
| ISSN (Print) | 0302-9743 |
| ISSN (Electronic) | 1611-3349 |
Other
| Other | 21st International Workshop on Combinatorial Algorithms, IWOCA 2010 |
|---|---|
| Country | United Kingdom |
| City | London |
| Period | 7/26/10 → 7/28/10 |
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All Science Journal Classification (ASJC) codes
- Computer Science(all)
- Theoretical Computer Science
Cite this
The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2. / Hasunuma, Toru; Ishii, Toshimasa; Ono, Hirotaka; Uno, Yushi.
Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, Revised Selected Papers. 2011. p. 103-106 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6460 LNCS).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
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 -