TY - GEN
T1 - The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2
AU - Hasunuma, Toru
AU - Ishii, Toshimasa
AU - Ono, Hirotaka
AU - Uno, Yushi
PY - 2011
Y1 - 2011
N2 - 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.
AB - 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.
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U2 - 10.1007/978-3-642-19222-7_11
DO - 10.1007/978-3-642-19222-7_11
M3 - Conference contribution
AN - SCOPUS:79953195503
SN - 9783642192210
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 103
EP - 106
BT - Combinatorial Algorithms - 21st International Workshop, IWOCA 2010, Revised Selected Papers
T2 - 21st International Workshop on Combinatorial Algorithms, IWOCA 2010
Y2 - 26 July 2010 through 28 July 2010
ER -