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

  • Authors:

  • Toru Hasunuma;Toshimasa Ishii;Hirotaka Ono;Yushi Uno

  • Affiliations:

  • Department of Mathematical and Natural Sciences, The University of Tokushima, Tokushima Japan;Department of Information and Management Science, Otaru University of Commerce, Otaru, Japan;Department of Economic Engineering, Kyushu University, Fukuoka, Japan;Department of Mathematics and Information Sciences, Graduate School of Science, Osaka Prefecture University, Sakai, Japan

  • Venue:
  • IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
  • Year:
  • 2010

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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 λ2T(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 λ2T(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 λ2T(G) ≤ Δ(G)+2 even in the case of Δ(G) ≤ 4.