683 research outputs found
Automated Discharging Arguments for Density Problems in Grids
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
and a lower bound of . 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 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
In this paper we present some results for a connected infinite graph with
finite degrees where the properties of balls of small radii guarantee the
existence of some Hamiltonian and connectivity properties of . (For a vertex
of a graph the ball of radius centered at is the subgraph of
induced by the set of vertices whose distance from does not
exceed ). In particular, we prove that if every ball of radius 2 in is
2-connected and satisfies the condition for
each path in , where and are non-adjacent vertices, then
has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017).
Furthermore, we prove that if every ball of radius 1 in satisfies Ore's
condition (1960) then all balls of any radius in are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio
Minimizing the regularity of maximal regular antichains of 2- and 3-sets
Let be a natural number. We study the problem to find the
smallest such that there is a family of 2-subsets and
3-subsets of with the following properties: (1)
is an antichain, i.e. no member of is a subset of
any other member of , (2) is maximal, i.e. for every
there is an with or , and (3) is -regular, i.e. every point
is contained in exactly members of . We prove lower
bounds on , 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
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
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
- …