7 research outputs found
Distance graphs with maximum chromatic number
Let be a finite set of integers. The distance graph has the set of integers as vertices and two vertices at distance are adjacent in . A conjecture of Xuding Zhu states that if the chromatic number of achieves its maximum value then the graph has a clique of order . We prove that the chromatic number of a distance graph with is five if and only if either or with and . This confirms Zhu's conjecture for
Packing Chromatic Number of Distance Graphs
The packing chromatic number of a graph is the smallest
integer such that vertices of can be partitioned into disjoint classes
where vertices in have pairwise distance greater than
. We study the packing chromatic number of infinite distance graphs , i.e. graphs with the set of integers as vertex set and in which two
distinct vertices are adjacent if and only if . In
this paper we focus on distance graphs with . We improve some
results of Togni who initiated the study. It is shown that for sufficiently large odd and
for sufficiently large even . We also give a lower bound 12 for
and tighten several gaps for with small .Comment: 13 pages, 3 figure
On Packing Colorings of Distance Graphs
The {\em packing chromatic number} of a graph is the
least integer for which there exists a mapping from to
such that any two vertices of color are at distance at
least . This paper studies the packing chromatic number of infinite
distance graphs , i.e. graphs with the set of
integers as vertex set, with two distinct vertices being
adjacent if and only if . We present lower and upper bounds for
, showing that for finite , the packing
chromatic number is finite. Our main result concerns distance graphs with
for which we prove some upper bounds on their packing chromatic
numbers, the smaller ones being for :
if is odd and
if is even
The lonely runner with seven runners
Suppose runners having nonzero constant speeds run laps on a
unit-length circular track starting at the same time and place. A runner is
said to be lonely if she is at distance at least along the track to
every other runner. The lonely runner conjecture states that every runner gets
lonely. The conjecture has been proved up to six runners (). A
formulation of the problem is related to the regular chromatic number of
distance graphs. We use a new tool developed in this context to solve the first
open case of the conjecture with seven runners
On the independence ratio of distance graphs
A distance graph is an undirected graph on the integers where two integers
are adjacent if their difference is in a prescribed distance set. The
independence ratio of a distance graph is the maximum density of an
independent set in . Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM
J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is
equal to the inverse of the fractional chromatic number, thus relating the
concept to the well studied question of finding the chromatic number of
distance graphs.
We prove that the independence ratio of a distance graph is achieved by a
periodic set, and we present a framework for discharging arguments to
demonstrate upper bounds on the independence ratio. With these tools, we
determine the exact independence ratio for several infinite families of
distance sets of size three, determine asymptotic values for others, and
present several conjectures.Comment: 39 pages, 12 figures, 6 table
Distance graphs with maximum chromatic number
Let D be a finite set of integers. The distance graph G(D) has the set of integers as vertices and two vertices at distance d ∈ D are adjacent in G(D). A conjecture of Xuding Zhu states that if the chromatic number of G(D) achieves its maximum value |D | + 1 then the graph has a clique of order |D|. We prove that the chromatic number of a distance graph with D = {a, b, c, d} is five if and only if either D = {1, 2, 3, 4k} or D = {a, b, a + b, a + 2b} with a ≡ 0 (mod 2) and b ≡ 1 (mod 2). This confirms Zhu’s conjecture for |D | = 4