839 research outputs found
Vertex arboricity of triangle-free graphs
Master's Project (M.S.) University of Alaska Fairbanks, 2016The vertex arboricity of a graph is the minimum number of colors needed to color the vertices so that the subgraph induced by each color class is a forest. In other words, the vertex arboricity of a graph is the fewest number of colors required in order to color a graph such that every cycle has at least two colors. Although not standard, we will refer to vertex arboricity simply as arboricity. In this paper, we discuss properties of chromatic number and k-defective chromatic number and how those properties relate to the arboricity of trianglefree graphs. In particular, we find bounds on the minimum order of a graph having arboricity three. Equivalently, we consider the largest possible vertex arboricity of triangle-free graphs of fixed order
Growth rates of geometric grid classes of permutations
Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the matching polynomial of a related graph. As a consequence, we characterise the set of growth rates of geometric grid classes in terms of the spectral radii of trees, explore the influence of "cycle parity" on the growth rate, compare the growth rates of geometric grid classes against those of the corresponding monotone grid classes, and present new results concerning the effect of edge subdivision on the largest root of the matching polynomial
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting
distribution for the number of fixed points for involutions chosen uniformly at
random from that class. In particular we consider monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
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
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