id: 05897203
dt: a
an: 05897203
au: Hasunuma, Toru; Ishii, Toshimasa; Ono, Hirotaka; Uno, Yushi
ti: The $(2,1)$-total labeling number of outerplanar graphs is at most $Δ+ 2$.
so: Iliopoulos, Costas S. (ed.) et al., Combinatorial algorithms. 21st
international workshop, IWOCA 2010, London, UK, July 26‒28, 2010.
Revised selected papers. Berlin: Springer (ISBN 978-3-642-19221-0/pbk).
Lecture Notes in Computer Science 6460, 103-106 (2011).
py: 2011
pu: Berlin: Springer
la: EN
cc:
ut:
ci: Zbl 1129.05041
li: doi:10.1007/978-3-642-19222-7_11
ab: Summary: 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 $\{0,1,\dots
,k\}$ of nonnegative integers such that $|f(x) - f(y)| \geq 2$ if $x$
is a vertex and $y$ is an edge incident to $x$, and $|f(x) - f(y)| \geq
1$ if $x$ and $y$ are a pair of adjacent vertices or a pair of adjacent
edges, for all $x$ and $y$ in $V(G) \cup E(G)$. The $(2,1)$-total
labeling number $λ^T_2(G)$ of $G$ is defined as the minimum $k$ among
all possible assignments. In [{\it D. Chen} and {\it W. Wang}, Discrete
Appl. Math. 155, No. 18, 2585‒2593 (2007; Zbl 1129.05041)], it was
conjectured that all outerplanar graphs $G$ satisfy $λ^T_2(G) \leq
Δ(G)+2$, where $Δ(G)$ is the maximum degree of $G$, while they also
showed that it is true for $G$ with $Δ(G) \geq 5$. In this paper, we
solve their conjecture completely, by proving that $λ^T_2(G) \leq
Δ(G)+2$ even in the case of $Δ(G) \leq 4$ .
rv: