A tight upper bound on the (2,1)-total labeling number of outerplanar graphs

  • Authors:

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

  • Affiliations:

  • Department of Mathematical and Natural Sciences, The University of Tokushima, Tokushima 770-8502, Japan;Department of Information and Management Science, Otaru University of Commerce, Otaru 047-8501, Japan;Department of Economic Engineering, Kyushu University, Fukuoka 812-8581, Japan;Department of Mathematics and Information Sciences, Graduate School of Science, Osaka Prefecture University, Sakai 599-8531, Japan

  • Venue:
  • Journal of Discrete Algorithms
  • Year:
  • 2012

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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 @l"2^T(G) of G is defined as the minimum k among all possible (2,1)-total labelings of G. In 2007, Chen and Wang conjectured that all outerplanar graphs G satisfy @l"2^T(G)==5. In this paper, we solve their conjecture, by proving that @l"2^T(G)=