115 research outputs found

    The inapproximability for the (0,1)-additive number

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    An {\it additive labeling} of a graph GG is a function :V(G)N \ell :V(G) \rightarrow\mathbb{N}, such that for every two adjacent vertices v v and u u of G G , wv(w)wu(w) \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) (xy x \sim y means that x x is joined to yy). The {\it additive number} of G G , denoted by η(G)\eta(G), is the minimum number kk such that G G has a additive labeling :V(G)Nk \ell :V(G) \rightarrow \mathbb{N}_k. The {\it additive choosability} of a graph GG, denoted by η(G)\eta_{\ell}(G) , is the smallest number kk such that GG has an additive labeling for any assignment of lists of size kk to the vertices of GG, such that the label of each vertex belongs to its own list. Seamone (2012) \cite{a80} conjectured that for every graph GG, η(G)=η(G)\eta(G)= \eta_{\ell}(G). We give a negative answer to this conjecture and we show that for every kk there is a graph GG such that η(G)η(G)k \eta_{\ell}(G)- \eta(G) \geq k. A {\it (0,1)(0,1)-additive labeling} of a graph GG is a function :V(G){0,1} \ell :V(G) \rightarrow\{0,1\}, such that for every two adjacent vertices v v and u u of G G , wv(w)wu(w) \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) . A graph may lack any (0,1)(0,1)-additive labeling. We show that it is NP \mathbf{NP} -complete to decide whether a (0,1)(0,1)-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph GG with some (0,1)(0,1)-additive labelings, the (0,1)(0,1)-additive number of GG is defined as σ1(G)=minΓvV(G)(v) \sigma_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v) where Γ\Gamma is the set of (0,1)(0,1)-additive labelings of GG. We prove that given a planar graph that admits a (0,1)(0,1)-additive labeling, for all ε>0 \varepsilon >0 , approximating the (0,1)(0,1)-additive number within n1ε n^{1-\varepsilon} is NP \mathbf{NP} -hard.Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer Scienc

    Sigma Partitioning: Complexity and Random Graphs

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    A sigma partitioning\textit{sigma partitioning} of a graph GG is a partition of the vertices into sets P1,,PkP_1, \ldots, P_k such that for every two adjacent vertices uu and vv there is an index ii such that uu and vv have different numbers of neighbors in PiP_i. The  sigma number\textit{ sigma number} of a graph GG, denoted by σ(G)\sigma(G), is the minimum number kk such that G G has a sigma partitioning P1,,PkP_1, \ldots, P_k. Also, a  lucky labeling\textit{ lucky labeling} of a graph GG is a function :V(G)N \ell :V(G) \rightarrow \mathbb{N}, such that for every two adjacent vertices v v and u u of G G , wv(w)wu(w) \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) (xy x \sim y means that x x and yy are adjacent). The  lucky number\textit{ lucky number} of G G , denoted by η(G)\eta(G), is the minimum number kk such that G G has a lucky labeling :V(G)Nk \ell :V(G) \rightarrow \mathbb{N}_k. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is NP \mathbf{NP} -complete to decide whether η(G)=2 \eta(G)=2 for a given 3-regular graph GG. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph

    Weight choosability of oriented hypergraphs

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    Master index to volumes 251-260

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    1-2-3 Conjecture in Digraphs: More Results and Directions

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    Horňak, Przybyło and Woźniak recently proved that almost every digraph can be 4-arc-weighted so that, for every arc u->v, the sum of weights incoming to u is different from the sum of weights outgoing from v. They conjectured a stronger result, namely that the same statement with 3 instead of 4 should also be true. We verify this conjecture in this work. This work takes place in a recent "quest" towards a directed version of the 1-2-3 Conjecture, the variant above being one of the last introduced ones. We take the occasion of this work to establish a summary of all results known in this field, covering known upper bounds, complexity aspects, and choosability. On the way we prove additional results which were missing in the whole picture. We also mention the aspects that remain open

    A characterization on orientations of graphs avoiding given lists on out-degrees

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    Let GG be a graph and F:V(G)2NF:V(G)\to2^N be a set function. The graph GG is said to be \emph{F-avoiding} if there exists an orientation OO of GG such that dO+(v)F(v)d^+_O(v)\notin F(v) for every vV(G)v\in V(G), where dO+(v)d^+_O(v) denotes the out-degree of vv in the directed graph GG with respect to OO. In this paper, we give a Tutte-type good characterization to decide the FF-avoiding problem when for every vV(G)v\in V(G), F(v)12(dG(v)+1)|F(v)|\leq \frac{1}{2}(d_G(v)+1) and F(v)F(v) contains no two consecutive integers. Our proof also gives a simple polynomial algorithm to find a desired orientation. As a corollary, we prove the following result: if for every vV(G)v\in V(G), F(v)12(dG(v)+1)|F(v)|\leq \frac{1}{2}(d_G(v)+1) and F(v)F(v) contains no two consecutive integers, then GG is FF-avoiding. This partly answers a problem proposed by Akbari et. al.(2020
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