28 research outputs found

    On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture

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    International audienceThis paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices

    Proof of a local antimagic conjecture

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    An antimagic labelling of a graph GG is a bijection f:E(G){1,,E(G)}f:E(G)\to\{1,\ldots,E(G)\} such that the sums Sv=evf(e)S_v=\sum_{e\ni v}f(e) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1994) is that every connected graph other than K2K_2 admits an antimagic labelling. Recently, two sets of authors (Arumugam, Premalatha, Ba\v{c}a \& Semani\v{c}ov\'a-Fe\v{n}ov\v{c}\'ikov\'a (2017), and Bensmail, Senhaji \& Lyngsie (2017)) independently introduced the weaker notion of a local antimagic labelling, where only adjacent vertices must be distinguished. Both sets of authors conjectured that any connected graph other than K2K_2 admits a local antimagic labelling. We prove this latter conjecture using the probabilistic method. Thus the parameter of local antimagic chromatic number, introduced by Arumugam et al., is well-defined for every connected graph other than K2K_2 .Comment: Final version for publication in DMTCS. Changes from previous version are formatting to journal style and correction of two minor typographical error

    Antimagic Labelings of Weighted and Oriented Graphs

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    A graph GG is kk-weightedlistantimagicweighted-list-antimagic if for any vertex weighting ω ⁣:V(G)R\omega\colon V(G)\to\mathbb{R} and any list assignment L ⁣:E(G)2RL\colon E(G)\to2^{\mathbb{R}} with L(e)E(G)+k|L(e)|\geq |E(G)|+k there exists an edge labeling ff such that f(e)L(e)f(e)\in L(e) for all eE(G)e\in E(G), labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on nn vertices having no K1K_1 or K2K_2 component is 4n3\lfloor{\frac{4n}{3}}\rfloor-weighted-list-antimagic. An oriented graph GG is kk-orientedantimagicoriented-antimagic if there exists an injective edge labeling from E(G)E(G) into {1,,E(G)+k}\{1,\dotsc,|E(G)|+k\} such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on nn vertices with no K1K_1 component admits an orientation that is 2n3\lfloor{\frac{2n}{3}}\rfloor-oriented-antimagic.Comment: 10 pages, 1 figur

    Regular graphs of odd degree are antimagic

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    An antimagic labeling of a graph GG with mm edges is a bijection from E(G)E(G) to {1,2,,m}\{1,2,\ldots,m\} such that for all vertices uu and vv, the sum of labels on edges incident to uu differs from that for edges incident to vv. Hartsfield and Ringel conjectured that every connected graph other than the single edge K2K_2 has an antimagic labeling. We prove this conjecture for regular graphs of odd degree.Comment: 5 page

    Combinatorial nullstellensatz and its applications

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    In 1999, Noga Alon proved a theorem, which he called the Combinatorial Nullstellensatz, that gives an upper bound to the number of zeros of a multivariate polynomial. The theorem has since seen heavy use in combinatorics, and more specifically in graph theory. In this thesis we will give an overview of the theorem, and of how it has since been applied by various researchers. Finally, we will provide an attempt at a proof utilizing a generalized version of the Combinatorial Nullstellensatz of the GM-MDS Conjecture

    Local Antimagic Coloring of Some Graphs

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    Given a graph G=(V,E)G =(V,E), a bijection f:E{1,2,,E}f: E \rightarrow \{1, 2, \dots,|E|\} is called a local antimagic labeling of GG if the vertex weight w(u)=uvEf(uv)w(u) = \sum_{uv \in E} f(uv) is distinct for all adjacent vertices. The vertex weights under the local antimagic labeling of GG induce a proper vertex coloring of a graph GG. The \textit{local antimagic chromatic number} of GG denoted by χla(G)\chi_{la}(G) is the minimum number of weights taken over all such local antimagic labelings of GG. In this paper, we investigate the local antimagic chromatic numbers of the union of some families of graphs, corona product of graphs, and necklace graph and we construct infinitely many graphs satisfying χla(G)=χ(G)\chi_{la}(G) = \chi(G)
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