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

Toru Hasunuma; Toshimasa Ishii; Hirotaka Ono; Yushi Uno

doi:10.1007/978-3-642-19222-7_11

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

Toru Hasunuma, Toshimasa Ishii, Hirotaka Ono, Yushi Uno

mathematics and information

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

2 Citations (Scopus)

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	https://doi.org/10.1007/978-3-642-19222-7_11
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

All Science Journal Classification (ASJC) codes

Computer Science(all)
Theoretical Computer Science

Access to Document

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

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Hasunuma, T., Ishii, T., Ono, H., & Uno, Y. (2011). The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2. In 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.

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

Hasunuma, T, Ishii, T, Ono, H & Uno, Y 2011, The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2. in 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

Hasunuma T, Ishii T, Ono H, Uno Y. The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2. In 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)). https://doi.org/10.1007/978-3-642-19222-7_11

Hasunuma, Toru ; Ishii, Toshimasa ; Ono, Hirotaka ; Uno, Yushi. / The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2. Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, Revised Selected Papers. 2011. pp. 103-106 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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