Abstract
An <i>additive labeling</i> of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$ ($x \sim y$ means that $x$ is joined to $y$). The additive number of $G$, denoted by $\eta (G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell : V(G) \rightarrow \mathbb{N}_k$. The additive choosability of a graph $G$, denoted by $\eta_\ell (G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone in his PhD thesis conjectured that for every graph $G$, $\eta(G)= \eta_\ell (G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_\ell (G) - \eta(G) \geq k$. A $(0,1)$-<i>additive labeling</i> of a graph $G$ is a function $\ell :V(G) \rightarrow \{0,1 \}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is NP-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_1 (G) = \mathrm{min}_{\ell \in \Gamma} \Sigma_{v \in V (G)} \ell (v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\epsilon > 0$ , approximating the $(0,1)$-additive number within $n^{1-\epsilon}$ is NP-hard.
Highlights
Throughout the paper we denote {1, 2, . . . , k} by Nk
An additive labeling of a graph G, which was introduced by Czerwinski et al [11], is a function : V (G) → N, such that for every two adjacent vertices v and u of G, w∼v (w) = w∼u (w) (x ∼ y means that x is joined to y)
We prove that given a planar graph that admits a (0, 1)-additive labeling, for all ε > 0, approximating the (0, 1)-additive number within n1−ε is NP-hard
Summary
Throughout the paper we denote {1, 2, . . . , k} by Nk. An additive labeling of a graph G, which was introduced by Czerwinski et al [11], is a function : V (G) → N, such that for every two adjacent vertices v and u of G, w∼v (w) = w∼u (w) (x ∼ y means that x is joined to y). Additive labeling and sigma coloring have been studied extensively by several authors, for instance see [3, 4, 6, 8, 10, 11, 13, 21, 22] It is proved, in [3] that it is NP-complete to determine whether a given graph G has η(G) = k for any k ≥ 2. Theorem 3 The following problem is NP-complete: Given a perfect graph G, does G have any (0, 1)additive labeling?. For a given graph G with a (0, 1)-additive labeling the function f (v) = 1 + w∼v (w) is a proper vertex coloring, so we have the following trivial lower bound for σ1(G).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
More From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.