2009/11/24 by Toru Hasunuma, Toshimasa Ishii, Hasunuma, Toru +5
Computer Science · Engineering · #Advanced Graph Theory Research #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Graph Labeling and Dimension Problems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.0911.4590
openalex publication_date 2009/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 λT2(G) of a graph 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, 2585--2593 (2007)], Chen and Wang conjectured that all outerplanar graphs G satisfy λT2(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 λT2(G) ≤ Δ(G)+2 even in the case of Δ(G)≤ 4 .