7 research outputs found
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
Packing Coloring of Undirected and Oriented Generalized Theta Graphs
The packing chromatic number (G) of an undirected (resp.
oriented) graph G is the smallest integer k such that its set of vertices V (G)
can be partitioned into k disjoint subsets V 1,..., V k, in such a way that
every two distinct vertices in V i are at distance (resp. directed distance)
greater than i in G for every i, 1 i k. The generalized theta graph
{\ell} 1,...,{\ell}p consists in two end-vertices joined by p 2
internally vertex-disjoint paths with respective lengths 1 {\ell} 1
. . . {\ell} p. We prove that the packing chromatic number of any
undirected generalized theta graph lies between 3 and max{5, n 3 + 2}, where n
3 = |{i / 1 i p, {\ell} i = 3}|, and that both these bounds are
tight. We then characterize undirected generalized theta graphs with packing
chromatic number k for every k 3. We also prove that the packing
chromatic number of any oriented generalized theta graph lies between 2 and 5
and that both these bounds are tight.Comment: Revised version. Accepted for publication in Australas. J. Combi
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
Packing Chromatic Number of Distance Graphs
The packing chromatic number χρ(G) of a graph G is the smallest integer k such that vertices of G can be partitioned into disjoint classes X1,..., Xk where vertices in Xi have pairwise distance greater than i. We study the packing chromatic number of infinite distance graphs G(Z, D), i.e. graphs with the set Z of integers as vertex set and in which two distinct vertices i, j ∈ Z are adjacent if and only if |i − j | ∈ D. In this paper we focus on distance graphs with D = {1, t}. We improve some results of Togni who initiated the study. It is shown that χρ(G(Z, D)) ≤ 35 for sufficiently large odd t and χρ(G(Z, D)) ≤ 56 for sufficiently large even t. We also give a lower bound 12 for t ≥ 9 and tighten several gaps for χρ(G(Z, D)) with small t