IWOCA 2010: Combinatorial Algorithms pp 103-106

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

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

## 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 $$\lambda^T_2(G)$$ 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 .

## References

1. 1.
Bazzaro, F., Montassier, M., Raspaud, A.: (d,1)-total labelling of planar graphs with large girth and high maximum degree. Discr. Math. 307, 2141–2151 (2007)
2. 2.
Behzad, M.: Graphs and their chromatic numbers. Ph.D. Thesis, Michigan State University (1965)Google Scholar
3. 3.
Calamoneri, T.: The L(h,k)-labelling problem: A survey and annotated bibliography. The Computer Journal 49, 585–608 (2006) (Updated version) (ver. October 19, 2009), http://www.dsi.uniroma1.it/~calamo/PDF-FILES/survey.pdf
4. 4.
Chen, D., Wang, W.: (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155, 2585–2593 (2007)
5. 5.
Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM J. Disc. Math. 5, 586–595 (1992)
6. 6.
Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A tight upper bound on the (2,1)-total labeling number of outerplanar graphs. CoRR abs/0911.4590 (2009)Google Scholar
7. 7.
Havet, F., Yu, M.-L.: (d,1)-Total labelling of graphs. Technical Report 4650, INRIA (2002)Google Scholar
8. 8.
Havet, F., Yu, M.-L.: (p,1)-Total labelling of graphs. Discr. Math. 308, 496–513 (2008)
9. 9.
Lih, K.-W., Liu, D.D.-F., Wang, W.: On (d,1)-total numbers of graphs. Discr. Math. 309, 3767–3773 (2009)
10. 10.
Montassier, M., Raspaud, A.: (d,1)-total labeling of graphs with a given maximum average degree. J. Graph Theory 51, 93–109 (2006)
11. 11.
Vizing, V.G.: Some unsolved problems in graph theory. Russian Mathematical Surveys 23, 125–141 (1968)
12. 12.
Wang, W., Zhang, K.: Δ-Matchings and edge-face chromatic numbers. Acta. Math. Appl. Sinica 22, 236–242 (1999)
13. 13.
Whittlesey, M.A., Georges, J.P., Mauro, D.W.: On the λ- number of Qn and related graphs. SIAM J. Discr. Math. 8, 499–506 (1995)
14. 14.
Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discr. Math. 306, 1217–1231 (2006)

## 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