The (2,1)-Total Labeling Number of Outerplanar Graphs Is at Most Δ + 2

  • Toru Hasunuma
  • Toshimasa Ishii
  • Hirotaka Ono
  • Yushi Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6460)


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 \(\lambda^T_2(G)\) 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 \(\lambda^T_2(G) \leq \Delta(G)+2\), 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 \(\lambda^T_2(G) \leq \Delta(G)+2\) even in the case of Δ(G) ≤ 4 .


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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Toru Hasunuma
    • 1
  • Toshimasa Ishii
    • 2
  • Hirotaka Ono
    • 3
  • Yushi Uno
    • 4
  1. 1.Department of Mathematical and Natural SciencesThe University of TokushimaTokushimaJapan
  2. 2.Department of Information and Management ScienceOtaru University of CommerceOtaruJapan
  3. 3.Department of Economic EngineeringKyushu UniversityFukuokaJapan
  4. 4.Department of Mathematics and Information Sciences, Graduate School of ScienceOsaka Prefecture UniversitySakaiJapan

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