683 research outputs found

    Automated Discharging Arguments for Density Problems in Grids

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    Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an identifying code in the hexagonal grid remains open, with an upper bound of 37≈0.428571\frac{3}{7} \approx 0.428571 and a lower bound of 512≈0.416666\frac{5}{12}\approx 0.416666. We present a new, experimental framework for producing discharging arguments using an algorithm. This algorithm replaces the lengthy case analysis of human-written discharging arguments with a linear program that produces the best possible lower bound using the specified set of discharging rules. We use this framework to present a lower bound of 2355≈0.418181\frac{23}{55} \approx 0.418181 on the density of an identifying code in the hexagonal grid, and also find several sharp lower bounds for variations on identifying codes in the hexagonal, square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables, and 2 figure

    Some local--global phenomena in locally finite graphs

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    In this paper we present some results for a connected infinite graph GG with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of GG. (For a vertex ww of a graph GG the ball of radius rr centered at ww is the subgraph of GG induced by the set Mr(w)M_r(w) of vertices whose distance from ww does not exceed rr). In particular, we prove that if every ball of radius 2 in GG is 2-connected and GG satisfies the condition dG(u)+dG(v)≥∣M2(w)∣−1d_G(u)+d_G(v)\geq |M_2(w)|-1 for each path uwvuwv in GG, where uu and vv are non-adjacent vertices, then GG has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in GG satisfies Ore's condition (1960) then all balls of any radius in GG are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio

    Minimizing the regularity of maximal regular antichains of 2- and 3-sets

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    Let n⩾3n\geqslant 3 be a natural number. We study the problem to find the smallest rr such that there is a family A\mathcal{A} of 2-subsets and 3-subsets of [n]={1,2,...,n}[n]=\{1,2,...,n\} with the following properties: (1) A\mathcal{A} is an antichain, i.e. no member of A\mathcal A is a subset of any other member of A\mathcal A, (2) A\mathcal A is maximal, i.e. for every X∈2[n]∖AX\in 2^{[n]}\setminus\mathcal A there is an A∈AA\in\mathcal A with X⊆AX\subseteq A or A⊆XA\subseteq X, and (3) A\mathcal A is rr-regular, i.e. every point x∈[n]x\in[n] is contained in exactly rr members of A\mathcal A. We prove lower bounds on rr, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference

    THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

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    and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

    Revisiting path-type covering and partitioning problems

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    This is a survey article which is at the initial stage. The author will appreciate to receive your comments and contributions to improve the quality of the article. The author's contact address is [email protected] problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts
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