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Abstract: A $2,1$-total labeling of a graph $G$ is an assignment $f$ from the vertexset $VG$ and the edge set $EG$ to the set $\{0,1,

.,k\}$ of nonnegativeintegers such that $|fx-fy|\ge 2$ if $x$ is a vertex and $y$ is an edgeincident to $x$, and $|fx-fy|\ge 1$ if $x$ and $y$ are a pair of adjacentvertices or a pair of adjacent edges, for all $x$ and $y$ in $VG\cup EG$.The $2,1$-total labeling number $\lambda^T 2G$ of a graph $G$ is defined asthe minimum $k$ among all possible assignments. In D. Chen and W. Wang.2,1-Total labelling of outerplanar graphs. Discr. Appl. Math. 155, 2585-25932007, Chen and Wang conjectured that all outerplanar graphs $G$ satisfy$\lambda^T 2G \leq \DeltaG+2$, where $\DeltaG$ is the maximum degree of$G$, while they also showed that it is true for $G$ with $\DeltaG\geq 5$. Inthis paper, we solve their conjecture completely, by proving that$\lambda^T 2G \leq \DeltaG+2$ even in the case of $\DeltaG\leq 4 $.



Author: Toru Hasunuma, Toshimasa Ishii, Hirotaka Ono, Yushi Uno

Source: https://arxiv.org/



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