11,594 research outputs found
Integer colorings with forbidden rainbow sums
For a set of positive integers , an -coloring of is
rainbow sum-free if it contains no rainbow Schur triple. In this paper we
initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of
sum-free sets, which asks for the subsets of with the maximum number of
rainbow sum-free -colorings. We show that for , the interval is
optimal, while for , the set is optimal. We
also prove a stability theorem for . The proofs rely on the hypergraph
container method, and some ad-hoc stability analysis.Comment: 20 page
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
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
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